
                     Fractint Version xx.xx                     Page 1

        New Features in Version 19.2...........................4
        Introduction...........................................9

  1.    Fractint Commands.....................................11
  1.1     Getting Started.....................................11
  1.2     Plotting Commands...................................11
  1.3     Zoom box Commands...................................14
  1.4     Color Cycling Commands..............................16
  1.5     Palette Editing Commands............................18
  1.6     Image Save/Restore Commands.........................22
  1.7     Print Command.......................................22
  1.8     Parameter Save/Restore Commands.....................23
  1.9     "3D" Commands.......................................25
  1.10    Interrupting and Resuming...........................25
  1.11    Orbits Window.......................................26
  1.12    View Window.........................................26
  1.13    Video Mode Function Keys............................28
  1.14    Browse Commands.....................................28
  1.15    RDS Commands........................................30
  1.16    Hints...............................................30
  1.17    Fractint on Unix....................................31
  2.    Fractal Types.........................................33
  2.1     The Mandelbrot Set..................................33
  2.2     Julia Sets..........................................34
  2.3     Julia Toggle Spacebar Commands......................35
  2.4     Inverse Julias......................................36
  2.5     Newton domains of attraction........................37
  2.6     Newton..............................................38
  2.7     Complex Newton......................................38
  2.8     Lambda Sets.........................................39
  2.9     Mandellambda Sets...................................39
  2.10    Circle..............................................40
  2.11    Plasma Clouds.......................................40
  2.12    Lambdafn............................................41
  2.13    Mandelfn............................................42
  2.14    Barnsley Mandelbrot/Julia Sets......................42
  2.15    Barnsley IFS Fractals...............................43
  2.16    Sierpinski Gasket...................................44
  2.17    Quartic Mandelbrot/Julia............................45
  2.18    Distance Estimator..................................45
  2.19    Pickover Mandelbrot/Julia Types.....................45
  2.20    Pickover Popcorn....................................46
  2.21    Peterson Variations.................................46
  2.22    Unity...............................................47
  2.23    Scott Taylor / Lee Skinner Variations...............47
  2.24    Kam Torus...........................................48
  2.25    Bifurcation.........................................48
  2.26    Orbit Fractals......................................50
  2.27    Lorenz Attractors...................................51
  2.28    Rossler Attractors..................................52
  2.29    Henon Attractors....................................52
  2.30    Pickover Attractors.................................53
  2.31    Gingerbreadman......................................53
  2.32    Martin Attractors...................................53
  2.33    Icon................................................54
  2.34    Test................................................54

                     Fractint Version xx.xx                     Page 2

  2.35    Formula.............................................55
  2.36    Julibrots...........................................57
  2.37    Diffusion Limited Aggregation.......................58
  2.38    Magnetic Fractals...................................59
  2.39    L-Systems...........................................60
  2.40    Lyapunov Fractals...................................62
  2.41    fn||fn Fractals.....................................63
  2.42    Halley..............................................63
  2.43    Dynamic System......................................64
  2.44    Mandelcloud.........................................65
  2.45    Quaternion..........................................65
  2.46    HyperComplex........................................66
  2.47    Cellular Automata...................................66
  2.48    Ant Automaton.......................................67
  2.49    Phoenix.............................................68
  2.50    Frothy Basins.......................................69

  3.    Doodads, Bells, and Whistles..........................71
  3.1     Drawing Method......................................71
  3.2     Palette Maps........................................72
  3.3     Autokey Mode........................................72
  3.4     Distance Estimator Method...........................74
  3.5     Inversion...........................................76
  3.6     Decomposition.......................................76
  3.7     Logarithmic Palettes and Color Ranges...............77
  3.8     Biomorphs...........................................78
  3.9     Continuous Potential................................79
  3.10    Starfields..........................................81
  3.11    Bailout Test........................................81
  3.12    Random Dot Stereograms (RDS)........................82

  4.    "3D" Images...........................................85
  4.1     3D Mode Selection...................................85
  4.2     Select Fill Type Screen.............................88
  4.3     Stereo 3D Viewing...................................89
  4.4     Rectangular Coordinate Transformation...............90
  4.5     3D Color Parameters.................................91
  4.6     Light Source Parameters.............................92
  4.7     Spherical Projection................................93
  4.8     3D Overlay Mode.....................................93
  4.9     Special Note for CGA or Hercules Users..............94
  4.10    Making Terrains.....................................94
  4.11    Making 3D Slides....................................96
  4.12    Interfacing with Ray Tracing Programs...............96

  5.    Command Line Parameters, Parameter Files, Batch Mode..99
  5.1     Using the DOS Command Line..........................99
  5.2     Setting Defaults (SSTOOLS.INI File).................99
  5.3     Parameter Files and the <@> Command................100
  5.4     General Parameter Syntax...........................101
  5.5     Startup Parameters.................................101
  5.6     Calculation Mode Parameters........................103
  5.7     Fractal Type Parameters............................103
  5.8     Image Calculation Parameters.......................104
  5.9     Color Parameters...................................106
  5.10    Doodad Parameters..................................109

                     Fractint Version xx.xx                     Page 3

  5.11    File Parameters....................................110
  5.12    Video Parameters...................................111
  5.13    Sound Parameters...................................113
  5.14    Printer Parameters.................................114
  5.15    PostScript Parameters..............................115
  5.16    PaintJet Parameters................................117
  5.17    Plotter Parameters.................................118
  5.18    3D Parameters......................................118
  5.19    Batch Mode.........................................120
  5.20    Browser Parameters.................................121

  6.    Hardware Support.....................................123
  6.1     Notes on Video Modes, "Standard" and Otherwise.....123
  6.2     "Disk-Video" Modes.................................125
  6.3     Customized Video Modes, FRACTINT.CFG...............126

  7.    Common Problems......................................129

  8.    Fractals and the PC..................................132
  8.1     A Little History...................................132
  8.1.1     Before Mandelbrot................................132
  8.1.2     Who Is This Guy, Anyway?.........................133
  8.2     A Little Code......................................134
  8.2.1     Periodicity Logic................................134
  8.2.2     Limitations of Integer Math (And How We Cope)....134
  8.2.3     Arbitrary Precision and Deep Zooming.............135
  8.2.4     The Fractint "Fractal Engine" Architecture.......137

  Appendix A Mathematics of the Fractal Types................139

  Appendix B Stone Soup With Pixels: The Authors.............158

  Appendix C GIF Save File Format............................166

  Appendix D Other Fractal Products..........................167

  Appendix E Bibliography....................................169

  Appendix F Other Programs..................................171

  Appendix G Revision History................................172

  Appendix H Version13 to Version 14 Type Mapping............190

                     Fractint Version xx.xx                     Page 4

 New Features in Version 19.2

  Version 19.2 is a bug-fix release for version 19.1. Changes from 19.1 to
    19.2 include:

  Fixed the 3D function, which was broken in 19.1 due to a side-effect of
    a repair of a minor bug in 19.0. Arrgghh! This is the main reason for
    the release of this version so quickly.

  Fixed a bug that caused the Julia inverse window and the orbits window
    to lose their place after loading a color map.

  Fixed a bug that causes corners to be lost when too many digits are
  entered.

  Added an enhanced ants automaton by Luciano Genero and Fulvio Cappelli.

  New showorbit command allows orbits-during-generation feature to be
    turned on by default. Expanded limits of Hertz command to 20 to 15000.

  Targa 3D files are now correctly written to workdir rather than tempdir.

  Uncommented garbage between file entries is now ignored. (But note that
    "{" must be on same line as entry name.)

  Fixed savename update logic.

  Version 19.1 is a bug-fix release for version 19.0. Changes from 19.0 to
    19.1 include:

  Disabled the F6 (corners) key when in the parameters screen (<z>) for
    arbitrary precision.

  IFS formulas now show in <z> screen.

  Allow RDS image maps of arbitrary dimensions.

  Touched up Mandelbrot/Julia <Space> toggle logic.

  Fractint now remembers map name, and uses the mapfile path correctly,
    and now allows periods in directory names.

  Fixed tab bug that caused problems when interrupting a restore of an
    arbitrary precision image.

  Repaired savename logic. No longer show (usually truncated) full path of
    the saved file in the screen.

  Fixed double to arbitrary precision transition with 90 degree images.
    (This only failed before when the image was rotated exactly 90
    degrees.)

  Corrected docs directory errors that reported several commands such as
    PARDIR= that were not implemented. Documented the color cycling HOME
    function.

                     Fractint Version xx.xx                     Page 5

  Fixed Mandelbrot/Julia types with bailout less than 4 (try it, results
    are interesting!)

  Fixed browser delete feature which left a box on the screen after
    deleting and exiting browser feature.

  More changes in filename processing logic.  ".\" is now recognized as
    the current directory and is expanded to its full path name.  It is
    now possible, although not recommended, to designate the root
    directory of a disk as the desired search directory.

  Fixed integer math Mandelbrot bug for 286 or lower machines.

  Fixed problem of reading some Lsys files incorrectly (distribution
    PENROSE.L file was broken unless first line was commented.)

  Fixed problem that caused endless loop in RDS with bad input values.

  Made reading the current directory first optional, added the new
    curdir=yes command for times when you want to use current directory
    files.

  Fixed problem with complexpower() function ("x^y" formula operator) in
    the case where x == 0. (Note that formulas where 0^0 appears for every
    every pixel are considered broken and no promises made.)

  Prevented aspect ratio drift as you zoom. If you want to make tiny
    adjustments, use new ASPECTDRIFT=0 command.

  Inside=bof60 and bof61 options now work correctly with the formula
  parser.

  We discovered the calculation time is no good after 24 days, so instead
    of the time you will now get the message "A Really Long Time!!! (>
    24.855 days)".  We thought you'd like to know ... A prize for the
    first person who actually *sees* this message!

  A summary of features new with 19.0 begins on next page.

                     Fractint Version xx.xx                     Page 6

  New arbitrary precision math allows types mandelbrot, julia, manzpower,
    and julzpower to zoom to 10^1600. See Arbitrary Precision and Deep
    Zooming (p. 135)

  New Random Dot Stereogram feature using <Ctrl>-<S>. Thanks to Paul De
    Leeuw for contributing this feature. For more, see Random Dot
    Stereograms (RDS) (p. 82).

  New browser invoked by the <l> command allows you to see the
    relationships of a family of images within the current corners values.
    See Browse Commands (p. 28) and Browser Parameters (p. 121). Thanks
    to Robin Bussell for contributing this feature.

  Added four Bailout Test (p. 81)s, real, imag, or, and.  These are set
    on the <Z> screen of the fractal types for which they work.  The
    default is still mod.

  New asin, asinh, acos, acosh, atan, atanh, sqrt, abs (abs(x)+i*abs(y)),
    and cabs (sqrt(x*x + y*y)) functions added to function variables and
    parser.

  New fractal types types chip, quadruptwo, threeply, phoenixcplx,
    mandphoenixclx, and ant automaton.

  Increased maximum iterations to 2,147,483,647 and maximum bailout to
     2,100,000,000 when using floating point math.

  New path/directory management. Fractint now remembers the pathname of
    command-line filenames. This means that you can specifiy directories
    where your files reside in SSTOOLS.INI. In what follows, <path> can be
    a directory, a filename, or a full path.

    File                      SSTOOLS.INI Command       Comments

    ==========================================================================
    PAR directory             parmfile=<path>
    GIF files for reading     filename=<path>
    MAP files                 map=<path>
    Autokey files             autokeyname=<path>
    GIF files for saving      savename=<path>
    Print file                printfile=<path>
    Formula files             formulafile=<path>
    Lsystem file              lfile=<path>
    IFS file                  ifsfile=<path>
    Miscellaneous files       workdir=<path>           new command
    Temporary files           tempdir=<path>           new command

    If the directories do not exist, Fractint gives an error message on
    runup with the option to continue.

                     Fractint Version xx.xx                     Page 7

  Fractint now searches all FRM, IFS, LSYS, and PAR files in the
    designated directory for entries. The number of entries in files has
    been greatly increased from 200 to 2000. Comment support in these
    files is improved.

  Parameters shown in <z> screen now match those used in a formula.

  Distance estimator logic has been overhauled, with the variable
    olddemmcolors added for backward compatibility.

  New floating point code for Lsystems from Nick Wilt greatly speeds up
     image generation.

  Enhanced fast parser from Chuck Ebbert makes floating point formula
    fractals faster than built-in types.

  Enhanced the history command to include all parameters, colors, and even
    .frm, .l, and .ifs file names and entries. Number of history sets
    remembered can be set with the maxhistory=<nnn> command to save
    memory.

  Enhanced center-mag coordinates to support rotated/stretched/skewed zoom
    boxes.

  Added new parameter to built-in Halley for comparison with formula type,
    also added new parameter to Frothybasin type.

  Added color number to orbits numbers <n> display.

  Added two new parameters to distest= to allow specifying resolution.
    This allows making resolution-independent distance estimator images.

  Fixed bug that caused the "big red switch" bug if '(' appeared in random
    uncommented formula file text, but fair warning, we don't officially
    support uncommented text in FRM files.

  Symmetry now works for the Marksjulia type and Marksmandel types.

  Full path no longer written in PAR files with <b> command.

  Fixed fractal type fn(z*z) so that zooming out will no longer dump you
    out to DOS, affecting zoomed out integer images made with this type.

  Fixed a float to fudged integer conversion that affects integer fractal
    types fn(z*z) and fn*fn.  This has only a minor impact on integer
    images made with these types.

  Default drive and directory restored after dropping to DOS, in case you
    changed it while under DOS.

  Added support for inversion to the formula parser (type=formula).

  Increased maximum number of files listed by <r> command to 2977 from
  300.

                     Fractint Version xx.xx                     Page 8

  Added outside=atan option.

  Added faster auto logmap logic.


                     Fractint Version xx.xx                     Page 9

 Introduction

  FRACTINT plots and manipulates images of "objects" -- actually, sets of
  mathematical points -- that have fractal dimension.  See "Fractals and
  the PC" (p. 132) for some historical and mathematical background on
  fractal geometry, a discipline named and popularized by mathematician
  Benoit Mandelbrot. For now, these sets of points have three important
  properties:

  1) They are generated by relatively simple calculations repeated over
  and over, feeding the results of each step back into the next --
  something computers can do very rapidly.

  2) They are, quite literally, infinitely complex: they reveal more and
  more detail without limit as you plot smaller and smaller areas.
  Fractint lets you "zoom in" by positioning a small box and hitting
  <Enter> to redraw the boxed area at full-screen size; its maximum linear
  "magnification" is over a trillionfold.

  3) They can be astonishingly beautiful, especially using PC color
  displays' ability to assign colors to selected points, and (with VGA
  displays or EGA in 640x350x16 mode) to "animate" the images by quickly
  shifting those color assignments.

  For a demonstration of some of Fractint's features, run the
  demonstration file included with this release (DEMO.BAT) by typing
  "demo" at the DOS prompt. You can stop the demonstration at any time by
  pressing <Esc>.

  The name FRACTINT was chosen because the program generates many of its
  images using INTeger math, rather than the floating point calculations
  used by most such programs. That means that you don't need a math co-
  processor chip (aka floating point unit or FPU), although for a few
  fractal types where floating point math is faster, the program
  recognizes and automatically uses an 80x87 chip if it's present. It's
  even faster on systems using Intel's 80386 and 80486 microprocessors,
  where the integer math can be executed in their native 32-bit mode.

  Fractint works with many adapters and graphics modes from CGA to the
  1024x768, 256-color XGA mode. Even "larger" images, up to 2048x2048x256,
  can be plotted to expanded memory, extended memory, or disk: this
  bypasses the screen and allows you to create images with higher
  resolution than your current display can handle, and to run in
  "background" under multi-tasking control programs such as DESQview and
  Windows 3.

  Fractint is an experiment in collaboration. Many volunteers have joined
  Bert Tyler, the program's first author, in improving successive
  versions.  Through electronic mail messages, CompuServe's GO GRAPHICS
  forums, new versions are hacked out and debugged a little at a time.
  Fractint was born fast, and none of us has seen any other fractal
  plotter close to the present version for speed, versatility, and all-
  around wonderfulness. (If you have, tell us so we can steal somebody
  else's ideas instead of each other's.)  See The Stone Soup Story
  (p. 158) and A Word About the Authors (p. 159) for information about the
  authors, and see Contacting the Authors (p. 161) for how to contribute

                     Fractint Version xx.xx                     Page 10

  your own ideas and code.

  Fractint is freeware. The copyright is retained by the Stone Soup Group.

  Fractint may be freely copied and distributed in unmodified form but may
  not be sold. (A nominal distribution fee may be charged for media and
  handling by freeware and shareware distributors.) Fractint may be used
  personally or in a business - if you can do your job better by using
  Fractint, or using images from it, that's great! It may not be given
  away with commercial products without explicit permission from the Stone
  Soup Group.

  There is no warranty of Fractint's suitability for any purpose, nor any
  acceptance of liability, express or implied.

   **********************************************************************
   * Contribution policy: Don't want money. Got money. Want admiration. *
   **********************************************************************

  Source code for Fractint is also freely available - see Distribution of
  Fractint (p. 160).  See the FRACTSRC.DOC file included with the source
  for conditions on use.  (In most cases we just want credit.)

                     Fractint Version xx.xx                     Page 11

 1. Fractint Commands


 1.1 Getting Started

  To start the program, enter FRACTINT at the DOS prompt. The program
  displays an initial "credits" screen. If Fractint doesn't start
  properly, please see Common Problems (p. 129).

  Hitting <Enter> gets you from the initial screen to the main menu. You
  can select options from the menu by moving the highlight with the cursor
  arrow keys and pressing <Enter>, or you can enter commands directly.

  As soon as you select a video mode, Fractint begins drawing an image -
  the "full" Mandelbrot set if you haven't selected another fractal type.

  For a quick start, after starting Fractint try one of the following:
    If you have MCGA, VGA, or better:  <F3>
    If you have EGA:                   <F9>
    If you have CGA:                   <F5>
    Otherwise, monochrome:             <F6>

  After the initial Mandelbrot image has been displayed, try zooming into
  it (see Zoom Box Commands (p. 14)) and color cycling (see Color Cycling
  Commands (p. 16)).  Once you're comfortable with these basics, start
  exploring other functions from the main menu.

  Help is available from the menu and at most other points in Fractint by
  pressing the <F1> key.

  AT ANY TIME, you can hit a command key to select a function. You do not
  need to wait for a calculation to finish, nor do you have to return to
  the main menu.

  When entering commands, note that for the "typewriter" keys, upper and
  lower case are equivalent, e.g. <B> and <b> have the same result.

  Many commands and parameters can be passed to FRACTINT as command-line
  arguments or read from a configuration file; see "Command Line
  Parameters, Parameter Files, Batch Mode" for details.


 1.2 Plotting Commands

  Function keys & various combinations are used to select a video mode and
  redraw the screen.  For a quick start try one of the following:
    If you have MCGA, VGA, or better:  <F3>
    If you have EGA:                   <F9>
    If you have CGA:                   <F5>
    Otherwise, monochrome:             <F6>

  <F1>
  Display a help screen. The function keys available in help mode are
  displayed at the bottom of the help screen.

                     Fractint Version xx.xx                     Page 12

  <M> or <Esc>
  Return from a displayed image to the main menu.

  <Esc>
  From the main menu, <Esc> is used to exit from Fractint.

  <Delete>
  Same as choosing "select video mode" from the main menu.  Goes to the
  "select video mode" screen.  See Video Mode Function Keys (p. 28).

  <h>
  Redraw the previous image in the circular history buffer, revisiting
  fractals you previously generated this session in reverse order.
  Fractint saves the last ten images worth of information including
  fractal type, coordinates, colors, and all options. Image information is
  saved only when some item changes. After ten images the circular buffer
  wraps around and earlier information is overwritten. You can set image
  capacity of the history feature using the maxhistory=<nnn> command.
  About 1200 bytes of memory is required for each image slot.

  <Ctrl-h>
  Redraw the next image in the circular history buffer. Use this to return
  to images you passed by when using <h>.

  <Tab>
  Display the current fractal type, parameters, video mode, screen or (if
  displayed) zoom-box coordinates, maximum iteration count, and other
  information useful in keeping track of where you are.  The Tab function
  is non-destructive - if you press it while in the midst of generating an
  image, you will continue generating it when you return.  The Tab
  function tells you if your image is still being generated or has
  finished - a handy feature for those overnight, 1024x768 resolution
  fractal images.  If the image is incomplete, it also tells you whether
  it can be interrupted and resumed.  (Any function other than <Tab> and
  <F1> counts as an "interrupt".)

  The Tab screen also includes a pixel-counting function, which will count
  the number of pixels colored in the inside color.  This gives an
  estimate of the area of the fractal.  Note that the inside color must be
  different from the outside color(s) for this to work; inside=0 is a good
  choice.

  <T>
  Select a fractal type. Move the cursor to your choice (or type the first
  few letters of its name) and hit <Enter>. Next you will be prompted for
  any parameters used by the selected type - hit <Enter> for the defaults.
  See Fractal Types (p. 33) for a list of supported types.

  <F>
  Toggles the use of floating-point algorithms (see "Limitations of
  Integer Math (And How We Cope)" (p. 134)).  Whether floating point is in
  use is shown on the <Tab> status screen.  The floating point option can
  also be turned on and off using the "X" options screen.  If you have a
  non-Intel floating point chip which supports the full 387 instruction
  set, see the "FPU=" command in Startup Parameters (p. 101) to get the
  most out of your chip.

                     Fractint Version xx.xx                     Page 13

  <X>
  Select a number of eXtended options. Brings up a full-screen menu of
  options, any of which you can change at will.  These options are:
    "passes=" - see Drawing Method (p. 71)
    Floating point toggle - see <F> key description below
    "maxiter=" - see Image Calculation Parameters (p. 104)
    "inside=" and "outside=" - see Color Parameters (p. 106)
    "savename=" filename - see File Parameters (p. 110)
    "overwrite=" option - see File Parameters (p. 110)
    "sound=" option - see Sound Parameters (p. 113)
    "logmap=" - see Logarithmic Palettes and Color Ranges (p. 77)
    "biomorph=" - see Biomorphs (p. 78)
    "decomp=" - see Decomposition (p. 76)
    "fillcolor=" - see Drawing Method (p. 71)

  <Y>
  More options which we couldn't fit under the <X> command:
    "finattract=" - see Finite Attractors (p. 152)
    "potential=" parameters - see Continuous Potential (p. 79)
    "invert=" parameters - see Inversion (p. 76)
    "distest=" parameters - see Distance Estimator Method (p. 74)
    "cyclerange=" - see Color Cycling Commands (p. 16)

  <Z>
  Modify the parameters specific to the currently selected fractal type.
  This command lets you modify the parameters which are requested when you
  select a new fractal type with the <T> command, without having to repeat
  that selection. You can enter "e" or "p" in column one of the input
  fields to get the numbers e and pi (2.71828... and 3.14159...).
  From the fractal parameters screen, you can press <F6> to bring up a sub
  parameter screen for the coordinates of the image's corners.  With
  selected fractal types, <Z> allows you to change the Bailout Test
  (p. 81).

  <+> or <->
  Switch to color-cycling mode and begin cycling the palette by shifting
  each color to the next "contour."  See Color Cycling Commands (p. 16).

  <C>
  Switch to color-cycling mode but do not start cycling.  The normally
  black "overscan" border of the screen changes to white.  See Color
  Cycling Commands (p. 16).

  <E>
  Enter Palette-Editing Mode.  See Palette Editing Commands (p. 18).

  <Spacebar>
  Toggle between Mandelbrot set images and their corresponding Julia-set
  images. Read the notes in Fractal Types, Julia Sets (p. 34) before
  trying this option if you want to see anything interesting.

  <J>
  Toggle between Julia escape time fractal and the Inverse Julia orbit
  fractal. See Inverse Julias (p. 36)

                     Fractint Version xx.xx                     Page 14

  <Enter>
  Enter is used to resume calculation after a pause. It is only necessary
  to do this when there is a message on the screen waiting to be
  acknowledged, such as the message shown after you save an image to disk.

  <I>
  Modify 3D transformation parameters used with 3D fractal types such as
  "Lorenz3D" and 3D "IFS" definitions, including the selection of "funny
  glasses" (p. 89) red/blue 3D.

  <A>
  Convert the current image into a fractal 'starfield'.  See Starfields
  (p. 81).

  <Ctrl-A>
  Unleash an image-eating ant automaton on current image. See Ant
  Automaton (p. 67).

  <Ctrl-S> (or <k>)
  Convert the current image into a Random Dot Stereogram (RDS).  See
  Random Dot Stereograms (RDS) (p. 82).

  <O> (the letter, not the number)
  If pressed while an image is being generated, toggles the display of
  intermediate results -- the "orbits" Fractint uses as it calculates
  values for each point. Slows the display a bit, but shows you how clever
  the program is behind the scenes. (See "A Little Code" in "Fractals and
  the PC" (p. 132).)

  <D>
  Shell to DOS. Return to Fractint by entering "exit" at a DOS prompt.

  <Insert>
  Restart at the "credits" screen and reset most variables to their
  initial state.  Variables which are not reset are: savename, lightname,
  video, startup filename.

  <L>
  Enter Browsing Mode.  See Browse Commands (p. 28).


 1.3 Zoom box Commands

  Zoom Box functions can be invoked while an image is being generated or
  when it has been completely drawn.  Zooming is supported for most
  fractal types, but not all.

  The general approach to using the zoom box is:  Frame an area using the
  keys described below, then <Enter> to expand what's in the frame to fill
  the whole screen (zoom in); or <Ctrl><Enter> to shrink the current image
  into the framed area (zoom out). With a mouse, double-click the left
  button to zoom in, double click the right button to zoom out.

  <Page Up>, <Page Down>
  Use <Page Up> to initially bring up the zoom box. It starts at full
  screen size. Subsequent use of these keys makes the zoom box smaller or

                     Fractint Version xx.xx                     Page 15

  larger.  Using <Page Down> to enlarge the zoom box when it is already at
  maximum size removes the zoom box from the display. Moving the mouse
  away from you or toward you while holding the left button down performs
  the same functions as these keys.

  Using the cursor "arrow" keys or moving the mouse without holding any
  buttons down, moves the zoom box.

  Holding <Ctrl> while pressing cursor "arrow" keys moves the box 5 times
  faster.  (This only works with enhanced keyboards.)

  Panning: If you move a fullsize zoombox and don't change anything else
  before performing the zoom, Fractint just moves what's already on the
  screen and then fills in the new edges, to reduce drawing time. This
  feature applies to most fractal types but not all.  A side effect is
  that while an image is incomplete, a full size zoom box moves in steps
  larger than one pixel.  Fractint keeps the box on multiple pixel
  boundaries, to make panning possible.  As a multi-pass (e.g. solid
  guessing) image approaches completion, the zoom box can move in smaller
  increments.

  In addition to resizing the zoom box and moving it around, you can do
  some rather warped things with it.  If you're a new Fractint user, we
  recommend skipping the rest of the zoom box functions for now and coming
  back to them when you're comfortable with the basic zoom box functions.

  <Ctrl><Keypad->, <Ctrl><Keypad+>
  Holding <Ctrl> and pressing the numeric keypad's + or - keys rotates the
  zoom box. Moving the mouse left or right while holding the right button
  down performs the same function.

  <Ctrl><Page Up>, <Ctrl><Page Down>
  These commands change the zoom box's "aspect ratio", stretching or
  shrinking it vertically. Moving the mouse away from you or toward you
  while holding both buttons (or the middle button on a 3-button mouse)
  down performs the same function. There are no commands to directly
  stretch or shrink the zoom box horizontally - the same effect can be
  achieved by combining vertical stretching and resizing.

  <Ctrl><Home>, <Ctrl><End>
  These commands "skew" the zoom box, moving the top and bottom edges in
  opposite directions. Moving the mouse left or right while holding both
  buttons (or the middle button on a 3-button mouse) down performs the
  same function. There are no commands to directly skew the left and right
  edges - the same effect can be achieved by using these functions
  combined with rotation.

  <Ctrl><Insert>, <Ctrl><Delete>
  These commands change the zoom box color. This is useful when you're
  having trouble seeing the zoom box against the colors around it. Moving
  the mouse away from you or toward you while holding the right button
  down performs the same function.

  You may find it difficult to figure out what combination of size,
  position rotation, stretch, and skew to use to get a particular result.
  (We do.)

                     Fractint Version xx.xx                     Page 16

  A good way to get a feel for all these functions is to play with the
  Gingerbreadman fractal type. Gingerbreadman's shape makes it easy to see
  what you're doing to him. A warning though: Gingerbreadman will run
  forever, he's never quite done! So, pre-empt with your next zoom when
  he's baked enough.

  If you accidentally change your zoom box shape or rotate and forget
  which way is up, just use <PageDown> to make it bigger until it
  disappears, then <PageUp> to get a fresh one.  With a mouse, after
  removing the old zoom box from the display release and re-press the left
  button for a fresh one.

  If your screen does not have a 4:3 "aspect ratio" (i.e. if the visible
  display area on it is not 1.333 times as wide as it is high), rotating
  and zooming will have some odd effects - angles will change, including
  the zoom box's shape itself, circles (if you are so lucky as to see any
  with a non-standard aspect ratio) become non-circular, and so on. The
  vast majority of PC screens *do* have a 4:3 aspect ratio.

  Zooming is not implemented for the plasma and diffusion fractal types,
  nor for overlayed and 3D images. A few fractal types support zooming but
  do not support rotation and skewing - nothing happens when you try it.


 1.4 Color Cycling Commands

  Color-cycling mode is entered with the 'c', '+', or '-' keys from an
  image, or with the 'c' key from Palette-Editing mode.

  The color-cycling commands are available ONLY for VGA adapters and EGA
  adapters in 640x350x16 mode.  You can also enter color-cycling while
  using a disk-video mode, to load or save a palette - other functions are
  not supported in disk-video.

  Note that the colors available on an EGA adapter (16 colors at a time
  out of a palette of 64) are limited compared to those of VGA, super-VGA,
  and MCGA (16 or 256 colors at a time out of a palette of 262,144). So
  color-cycling in general looks a LOT better in the latter modes. Also,
  because of the EGA palette restrictions, some commands are not available
  with EGA adapters.

  Color cycling applies to the color numbers selected by the "cyclerange="
  command line parameter (also changeable via the <Y> options screen and
  via the palette editor).  By default, color numbers 1 to 255 inclusive
  are cycled.  On some images you might want to set "inside=0" (<X>
  options or command line parameter) to exclude the "lake" from color
  cycling.

  When you are in color-cycling mode, you will either see the screen
  colors cycling, or will see a white "overscan" border when paused, as a
  reminder that you are still in this mode.  The keyboard commands
  available once you've entered color-cycling. are described below.

  <F1>
  Bring up a HELP screen with commands specific to color cycling mode.

                     Fractint Version xx.xx                     Page 17

  <Esc>
  Leave color-cycling mode.

  <Home>
  Restore original palette.

  <+> or <->
  Begin cycling the palette by shifting each color to the next "contour."
  <+> cycles the colors in one direction, <-> in the other.

  '<' or '>'
  Force a color-cycling pause, disable random colorizing, and single-step
  through a one color-cycle.  For "fine-tuning" your image colors.

  Cursor up/down
  Increase/decrease the cycling speed. High speeds may cause a harmless
  flicker at the top of the screen.

  <F2> through <F10>
  Switches from simple rotation to color selection using randomly
  generated color bands of short (F2) to long (F10) duration.

  <1> through <9>
  Causes the screen to be updated every 'n' color cycles (the default is
  1).  Handy for slower computers.

  <Enter>
  Randomly selects a function key (F2 through F10) and then updates ALL
  the screen colors prior to displaying them for instant, random colors.
  Hit this over and over again (we do).

  <Spacebar>
  Pause cycling with white overscan area. Cycling restarts with any
  command key (including another spacebar).

  <Shift><F1>-<F10>
  Pause cycling and reset the palette to a preset two color "straight"
  assignment, such as a spread from black to white. (Not for EGA)

  <Ctrl><F1>-<F10>
  Pause & set a 2-color cyclical assignment, e.g. red->yellow->red (not
  EGA).

  <Alt><F1>-<F10>
  Pause & set a 3-color cyclical assignment, e.g. green->white->blue (not
  EGA).

  <R>, <G>, <B>
  Pause and increase the red, green, or blue component of all colors by a
  small amount (not for EGA). Note the case distinction of this vs:

  <r>, <g>, <b>
  Pause and decrease the red, green, or blue component of all colors by a
  small amount (not for EGA).

                     Fractint Version xx.xx                     Page 18

  <D> or <A>
  Pause and load an external color map from the files DEFAULT.MAP or
  ALTERN.MAP, supplied with the program.

  <L>
  Pause and load an external color map (.MAP file).  Several .MAP files
  are supplied with Fractint.  See Palette Maps (p. 72).

  <S>
  Pause, prompt for a filename, and save the current palette to the named
  file (.MAP assumed).  See Palette Maps (p. 72).


 1.5 Palette Editing Commands

  Palette-editing mode provides a number of tools for modifying the colors
  in an image.  It can be used only with MCGA or higher adapters, and only
  with 16 or 256 color video modes.  Many thanks to Ethan Nagel for
  creating the palette editor.

  Use the <E> key to enter palette-editing mode from a displayed image or
  from the main menu.

  When this mode is entered, an empty palette frame is displayed. You can
  use the cursor keys to position the frame outline, and <Pageup> and
  <Pagedn> to change its size.  (The upper and lower limits on the size
  depend on the current video mode.)  When the frame is positioned where
  you want it, hit Enter to display the current palette in the frame.

  Note that the palette frame shows R(ed) G(reen) and B(lue) values for
  two color registers at the top.  The active color register has a solid
  frame, the inactive register's frame is dotted.  Within the active
  register, the active color component is framed.

  Using the commands described below, you can assign particular colors to
  the registers and manipulate them.  Note that at any given time there
  are two colors "X"d - these are pre-empted by the editor to display the
  palette frame. They can be edited but the results won't be visible. You
  can change which two colors are borrowed ("X"d out) by using the <v>
  command.

  Once the palette frame is displayed and filled in, the following
  commands are available:

  <F1>
  Bring up a HELP screen with commands specific to palette-editing mode.

  <Esc>
  Leave palette-editing mode

  <H>
  Hide the palette frame to see full image; the cross-hair remains visible
  and all functions remain enabled; hit <H> again to restore the palette
  display.

                     Fractint Version xx.xx                     Page 19

  Cursor keys
  Move the cross-hair cursor around. In 'auto' mode (the default) the
  color under the center of the cross-hair is automatically assigned to
  the active color register. Control-Cursor keys move the cross-hair
  faster. A mouse can also be used to move around.

  <R> <G> <B>
  Select the Red, Green, or Blue component of the active color register
  for subsequent commands

  <Insert> <Delete>
  Select previous or next color component in active register

  <+> <->
  Increase or decrease the active color component value by 1  Numeric
  keypad (gray) + and - keys do the same.

  <Pageup> <Pagedn>
  Increase or decrease the active color component value by 5; Moving the
  mouse up/down with left button held is the same

  <0> <1> <2> <3> <4> <5>
  Set the active color component's value to 0 10 20 ... 60

  <Space>
  Select the other color register as the active one.  In the default
  'auto' mode this results in the now-inactive register being set to
  remember the color under the cursor, and the now-active register
  changing from whatever it had previously remembered to now follow the
  color.

  <,> <.>
  Rotate the palette one step.  By default colors 1 through 255 inclusive
  are rotated.  This range can be over-ridden with the "cyclerange"
  parameter, the <Y> options screen, or the <O> command described below.

  "<" ">"
  Rotate the palette continuously (until next keystroke)

  <O>
  Set the color cycling range to the range of colors currently defined by
  the color registers.

  <C>
  Enter Color-Cycling Mode.  When you invoke color-cycling from here, it
  will subsequently return to palette-editing when you <Esc> from it.  See
  Color Cycling Commands (p. 16).

  <=>
  Create a smoothly shaded range of colors between the colors selected by
  the two color registers.

  <M>
  Specify a gamma value for the shading created by <=>.

                     Fractint Version xx.xx                     Page 20

  <D>
  Duplicate the inactive color register's values to the active color
  register.

  <T>
  Stripe-shade - create a smoothly shaded range of colors between the two
  color registers, setting only every Nth register.  After hitting <T>,
  hit a numeric key from 2 to 9 to specify N.  For example, if you press
  <T> <3>, smooth shading is done between the two color registers,
  affecting only every 3rd color between them.  The other colors between
  them remain unchanged.

  <W>
  Convert current palette to gray-scale.  (If the <X> or <Y> exclude
  ranges described later are in force, only the active range of colors is
  converted to gray-scale.)

  <Shift-F2> ... <Shift-F9>
  Store the current palette in a temporary save area associated with the
  function key.  The temporary save palettes are useful for quickly
  comparing different palettes or the effect of some changes - see next
  command.  The temporary palettes are only remembered until you exit from
  palette-editing mode.

  <F2> ... <F9>
  Restore the palette from a temporary save area.  If you haven't
  previously saved a palette for the function key, you'll get a simple
  grey scale.

  <L>
  Pause and load an external color map (.MAP file).  See Palette Maps
  (p. 72).

  <S>
  Pause, prompt for a filename, and save the current palette to the named
  file (.MAP assumed).  See Palette Maps (p. 72).

  <I>
  Invert frame colors.  With some colors the palette is easier to see when
  the frame colors are interchanged.
  <\>
  Move or resize the palette frame.  The frame outline is drawn - it can
  then be repositioned and sized with the cursor keys, <Pageup> and
  <Pagedn>, just as was done when first entering palette-editing mode.
  Hit Enter when done moving/sizing.

  <V>
  Use the colors currently selected by the two color registers for the
  palette editor's frame.  When palette editing mode is entered, the last
  two colors are "X"d out for use by the palette editor; this command can
  be used to replace the default with two other color numbers.

  <A>
  Toggle 'auto' mode on or off.  When on (the default), the active color
  register follows the cursor; when off, <Enter> must be pressed to set
  the active register to the color under the cursor.

                     Fractint Version xx.xx                     Page 21

  <Enter>
  Only useful when 'auto' is off, as described above; double clicking the
  left mouse button is the same as Enter.

  <X>
  Toggle 'exclude' mode on or off - when toggled on, only those image
  pixels which match the active color are displayed.

  <Y>
  Toggle 'exclude' range on or off - similar to <X>, but all pixels
  matching colors in the range of the two color registers are displayed.

  <N>
  Make a negative color palette - will convert only current color if in
  'x' mode or range between editors in 'y' mode or entire palette if in
  "normal" mode.

  <!>
  <@>
  <#>
  Swap R<->G, G<->B, and R<->B columns. These keys are shifted 1, 2, and
  3, which you may find easier to remember.

  <U>
  Undoes the last palette editor command.  Will undo all the way to the
  beginning of the current session.
  <E> Redoes the undone palette editor commands.

  <F>
  Toggles "Freestyle mode" on and off (Freestyle mode changes a range of
  palette values smoothly from a center value outward).  With your cursor
  inside the palette box, press the <F> key to enter Freestyle mode.  A
  default range of colors will be selected for you centered at the cursor
  (the ends of the color range are noted by putting dashed lines around
  the corresponding palette values). While in Freestyle mode:

   Moving the mouse changes the location of the range of colors that are
   affected.

   Control-Insert/Delete or the shifted-right-mouse-button changes the
   size of the affected palette range.

   The normal color editing keys (R,G,B,1-6, etc) set the central color of
   the affected palette range.
   Pressing ENTER or double-clicking the left mouse button makes the
   palette changes permanent (if you don't perform this step, any palette
   changes disappear when you press the <F> key again to exit freestyle
   mode).

                     Fractint Version xx.xx                     Page 22

 1.6 Image Save/Restore Commands

  <S> saves the current image to disk. All parameters required to recreate
  the image are saved with it. Progress is marked by colored lines moving
  down the screen's edges.

  The default filename for the first image saved after starting Fractint
  is FRACT001.GIF;  subsequent saves in the same session are automatically
  incremented 002, 003... Use the "savename=" parameter or <X> options
  screen to change the name. By default, files left over from previous
  sessions are not overwritten - the first unused FRACTnnn name is used.
  Use the "overwrite=yes" parameter or <X> options screen) to overwrite
  existing files.

  A save operation can be interrupted by pressing any key. If you
  interrupt, you'll be asked whether to keep or discard the partial file.

  <R> restores an image previously saved with <S>, or an ordinary GIF
  file.  After pressing <R> you are shown the file names in the current
  directory which match the current file mask. To select a file to
  restore, move the cursor to it (or type the first few letters of its
  name) and press <Enter>.

  Directories are shown in the file list with a "\" at the end of the
  name.  When you select a directory, the contents of that directory are
  shown. Or, you can type the name of a different directory (and
  optionally a different drive) and press <Enter> for a new display. You
  can also type a mask such as "*.XYZ" and press <Enter> to display files
  whose name ends with the matching suffix (XYZ).

  You can use <F6> to switch directories to the default fractint directory
  or to your own directory which is specified through the DOS environment
  variable "FRACTDIR".

  Once you have selected a file to restore, a summary description of the
  file is shown, with a video mode selection list. Usually you can just
  press <Enter> to go past this screen and load the image. Other choices
  available at this point are:
    Cursor keys: select a different video mode
    <Tab>: display more information about the fractal
    <F1>: for help about the "err" column in displayed video modes
  If you restore a file into a video mode which does not have the same
  pixel dimensions as the file, Fractint will make some adjustments:  The
  view window parameters (see <V> command) will automatically be set to an
  appropriate size, and if the image is larger than the screen dimensions,
  it will be reduced by using only every Nth pixel during the restore.


 1.7 Print Command

  <P>

  Print the current fractal image on your (Laserjet, Paintjet, Epson-
  compatible, PostScript, or HP-GL) printer.

                     Fractint Version xx.xx                     Page 23

  See "Setting Defaults (SSTOOLS.INI File)" (p. 99) and "Printer
  Parameters" (p. 114) for how to let Fractint know about your printer
  setup.

  "Disk-Video" Modes (p. 125) can be used to generate images for printing
  at higher resolutions than your screen supports.


 1.8 Parameter Save/Restore Commands

  Parameter files can be used to save/restore all options and settings
  required to recreate particular images.  The parameters required to
  describe an image require very little disk space, especially compared
  with saving the image itself.

  <@> or <2>

  The <@> or <2> command loads a set of parameters describing an image.
  (Actually, it can also be used to set non-image parameters such as
  SOUND, but at this point we're interested in images. Other uses of
  parameter files are discussed in "Parameter Files and the <@> Command"
  (p. 100).)

  When you hit <@> or <2>, Fractint displays the names of the entries in
  the currently selected parameter file.  The default parameter file,
  FRACTINT.PAR, is included with the Fractint release and contains
  parameters for some sample images.

  After pressing <@> or <2>, highlight an entry and press <Enter> to load
  it, or press <F6> to change to another parameter file.

  Note that parameter file entries specify all calculation related
  parameters, but do not specify things like the video mode - the image
  will be plotted in your currently selected mode.

  <B>

  The <B> command saves the parameters required to describe the currently
  displayed image, which can subsequently be used with the <@> or <2>
  command to recreate it.

  After you press <B>, Fractint prompts for:

    Parameter file:  The name of the file to store the parameters in.  You
    should use some name like "myimages" instead of fractint.par, so that
    your images are kept separate from the ones released with new versions
    of Fractint. You can use the PARMFILE= command in SSTOOLS.INI to set
    the default parameter file name to "myimages" or whatever.  (See
    "Setting Defaults (SSTOOLS.INI File)" (p. 99) and "parmfile=" in
    "File Parameters" (p. 110).)

    Name:  The name you want to assign to the entry, to be displayed when
    the <@> or <2> command is used.

                     Fractint Version xx.xx                     Page 24

    Main comment:  A comment to be shown beside the entry in the <@>
    command display.

    Second, Third, and Fourth comment:  Additional comments to store in
    the file with the entry. These comments go in the file only, and are
    not displayed by the <@> command.

    Record colors?:  Whether color information should be included in the
    entry. Usually the default value displayed by Fractint is what you
    want.  Allowed values are:
    "no" - Don't record colors. This is the default if the image is using
       your video adapter's default colors.
    "@mapfilename" - When these parameters are used, load colors from the
       named color map file. This is the default if you are currently
       using colors from a color map file.
    "yes" - Record the colors in detail. This is the default when you've
       changed the display colors by using the palette editor or by color
       cycling. The only reason that this isn't what Fractint always does
       for the <B> command is that color information can be bulky - up to
       nearly 1K of disk space. That may not sound like much, but can add
       up when you consider the thousands of wonderful images you may find
       you just *have* to record...  Smooth-shaded ranges of colors are
       compressed, so if that's used a lot in an image the color
       information won't be as bulky.

    # of colors:  This only matters if "Record colors?" is set to "yes".
    It specifies the number of colors to record. Recording less colors
    will take less space. Usually the default value displayed by Fractint
    is what you want. You might want to increase it in some cases, e.g. if
    you are using a 256 color mode with maxiter 150, and have used the
    palette editor to set all 256 possible colors for use with color
    cycling, then you'll want to set the "# of colors" to 256.

    At the bottom of the input screen are inputs for Fractint's "pieces"
    divide-and-conquer feature. You can create multiple PAR entries that
    break an image up into pieces so that you can generate the image
    pieces one by one. There are two reasons for doing this. The first is
    in case the fractal is very slow, and you want to generate parts of
    the image at the same time on several computers. The second is that
    you might want to make an image greater than 2048 x 2048. The
    parameters for this feature are:
       X Multiples - How many divisions of final image in the x direction
       Y Multiples - How many divisions of final image in the y direction
       Video mode  - Fractint video mode for each piece (e.g. "F3")

    The last item defaults to the current video mode. If either X
    Multiples or Y Multiples are greater than 1, then multiple numbered
    PAR entries for the pieces are added to the PAR file, and a
    MAKEMIG.BAT file is created that builds all of the component pieces
    and then stitches them together into a "multi-image" GIF.  The current
    limitations of the "divide and conquer" algorithm are 36 or fewer X
    and Y multiples (so you are limited to "only" 36x36=1296 component
    images), and a final resolution limit in both the X and Y directions
    of 65,535 (a limitation of "only" four billion pixels or so).

                     Fractint Version xx.xx                     Page 25

    The final image generated by MAKEMIG is a "multi-image" GIF file
    called FRACTMIG.GIF.  In case you have other software that can't
    handle multi-image GIF files, MAKEMIG includes a final (but commented
    out) call to SIMPLGIF, a companion program that reads a GIF file that
    may contain little tricks like multiple images and creates a simple
    GIF from it.  Fair warning: SIMPLGIF needs room to build a composite
    image while it works, and it does that using a temporary disk file
    equal to the size of the final image - and a 64Kx64K GIF image
    requires a 4GB temporary disk file!

  <G>

  The <G> command lets you give a startup parameter interactively.


 1.9 "3D" Commands

  See "3D" Images (p. 85) for details of these commands.

  <3>
  Restore a saved image as a 3D "landscape", translating its color
  information into "height". You will be prompted for all KINDS of
  options.

  <#>
  Restore in 3D and overlay the result on the current screen.


 1.10 Interrupting and Resuming

  Fractint command keys can be loosely grouped as:

   o Keys which suspend calculation of the current image (if one is being
     calculated) and automatically resume after the function.  <Tab>
     (display status information) and <F1> (display help), are the only
     keys in this group.

   o Keys which automatically trigger calculation of a new image.
     Examples:  selecting a video mode (e.g. <F3>);  selecting a fractal
     type using <T>;  using the <X> screen to change an option such as
     maximum iterations.

   o Keys which do something, then wait for you to indicate what to do
     next.  Examples:  <M> to go to main menu;  <C> to enter color cycling
     mode;  <PageUp> to bring up a zoom box.  After using a command in
     this group, calculation automatically resumes when you return from
     the function (e.g. <Esc> from color cycling, <PageDn> to clear zoom
     box).  There are a few fractal types which cannot resume calculation,
     they are noted below.  Note that after saving an image with <S>, you
     must press <Enter> to clear the "saved" message from the screen and
     resume.

  An image which is <S>aved before it completes can later be <R>estored
  and continued. The calculation is automatically resumed when you restore
  such an image.

                     Fractint Version xx.xx                     Page 26

  When a slow fractal type resumes after an interruption in the third
  category above, there may be a lag while nothing visible happens.  This
  is because most cases of resume restart at the beginning of a screen
  line.  If unsure, you can check whether calculation has resumed with the
  <Tab> key.

  The following fractal types cannot (currently) be resumed: plasma, 3d
  transformations, julibrot, and 3d orbital types like lorenz3d.  To check
  whether resuming an image is possible, use the <Tab> key while it is
  calculating.  It is resumable unless there is a note under the fractal
  type saying it is not.

  The Batch Mode (p. 120) section discusses how to resume in batch mode.

  To <R>estore and resume a "formula", "lsystem", or "ifs" type fractal
  your "formulafile", "lfile", or "ifsfile" must contain the required
  name.


 1.11 Orbits Window

  The <O> key turns on the Orbit mode.  In this mode a cursor appears over
  the fractal. A window appears showing the orbit used in the calculation
  of the color at the point where the cursor is. Move the cursor around
  the fractal using the arrow keys or the mouse and watch the orbits
  change. Try entering the Orbits mode with View Windows (<V>) turned on.
  The following keys take effect in Orbits mode.
  <c>         Circle toggle - makes little circles with radii inversely
              proportional to the iteration. Press <c> again to toggle
              back to point-by-point display of orbits.
  <l>         Line toggle - connects orbits with lines (can use with <c>)
  <n>         Numbers toggle - shows complex coordinates and color number
  of
              the cursor on the screen. Press <n> again to turn off numbers.
  <p>         Enter pixel coordinates directly
  <h>         Hide fractal toggle. Works only if View Windows is turned on
              and set for a small window (such as the default size.) Hides the
              fractal, allowing the orbit to take up the whole screen. Press
              <h> again to uncover the fractal.
  <s>         Saves the fractal, cursor, orbits, and numbers as they
  appear
              on the screen.
  <<> or <,>  Zoom orbits image smaller
  <>> or <.>  Zoom orbits image larger
  <z>         Restore default zoom.


 1.12 View Window

  The <V> command is used to set the view window parameters described
  below.  These parameters can be used to:
   o Define a small window on the screen which is to contain the generated
     images. Using a small window speeds up calculation time (there are
     fewer pixels to generate). You can use a small window to explore
     quickly, then turn the view window off to recalculate the image at
     full screen size.

                     Fractint Version xx.xx                     Page 27

   o Generate an image with a different "aspect ratio"; e.g. in a square
     window or in a tall skinny rectangle.
   o View saved GIF images which have pixel dimensions different from any
     mode supported by your hardware. This use of view windows occurs
     automatically when you restore such an image.

  "Preview display"
  Set this to "yes" to turn on view window, "no" for full screen display.
  While this is "no", the only view parameter which has any affect is
  "final media aspect ratio". When a view window is being used, all other
  Fractint functions continue to operate normally - you can zoom, color-
  cycle, and all the rest.

  "Reduction factor"
  When an explicit size is not given, this determines the view window
  size, as a factor of the screen size.  E.g. a reduction factor of 2
  makes the window 1/2 as big as the screen in both dimensions.

  "Final media aspect ratio"
  This is the height of the final image you want, divided by the width.
  The default is 0.75 because standard PC monitors have a height:width
  ratio of 3:4. E.g. set this to 2.0 for an image twice as high as it is
  wide. The effect of this parameter is visible only when "preview
  display" is enabled.

  "Crop starting coordinates"
  This parameter affects what happens when you change the aspect ratio. If
  set to "no", then when you change aspect ratio, the prior image will be
  squeezed or stretched to fit into the new shape. If set to "yes", the
  prior image is "cropped" to avoid squeezing or stretching.

  "Explicit size"
  Setting these to non-zero values over-rides the "reduction factor" with
  explicit sizes in pixels. If only the "x pixels" size is specified, the
  "y pixels" size is calculated automatically based on x and the aspect
  ratio.

  More about final aspect ratio:  If you want to produce a high quality
  hard-copy image which is say 8" high by 5" down, based on a vertical
  "slice" of an existing image, you could use a procedure like the
  following. You'll need some method of converting a GIF image to your
  final media (slide or whatever) - Fractint can only do the whole job
  with a PostScript printer, it does not preserve aspect ratio with other
  printers.
   o restore the existing image
   o set view parameters: preview to yes, reduction to anything (say 2),
     aspect ratio to 1.6, and crop to yes
   o zoom, rotate, whatever, till you get the desired final image
   o set preview display back to no
   o trigger final calculation in some high res disk video mode, using the
     appropriate video mode function key
   o print directly to a PostScript printer, or save the result as a GIF
     file and use external utilities to convert to hard copy.

                     Fractint Version xx.xx                     Page 28

 1.13 Video Mode Function Keys

  Fractint supports *so* many video modes that we've given up trying to
  reserve a keyboard combination for each of them.

  Any supported video mode can be selected by going to the "Select Video
  Mode" screen (from main menu or by using <Delete>), then using the
  cursor up and down arrow keys and/or <PageUp> and <PageDown> keys to
  highlight the desired mode, then pressing <Enter>.

  Up to 39 modes can be assigned to the keys F2-F10, SF1-SF10
  <Shift>+<Fn>), CF1-CF10 (<Ctrl>+<Fn>), and AF1-AF10 (<Alt>+<Fn>).  The
  modes assigned to function keys can be invoked directly by pressing the
  assigned key, without going to the video mode selection screen.

  30 key combinations can be reassigned:  <F1> to <F10> combined with any
  of <Shift>, <Ctrl>, or <Alt>.  The video modes assigned to <F2> through
  <F10> can not be changed - these are assigned to the most common video
  modes, which might be used in demonstration files or batches.

  To reassign a function key to a mode you often use, go to the "select
  video mode" screen, highlight the video mode, press the keypad (gray)
  <+> key, then press the desired function key or key combination.  The
  new key assignment will be remembered for future runs.

  To unassign a key (so that it doesn't invoke any video mode), highlight
  the mode currently selected by the key and press the keypad (gray) <->
  key.

  A note about the "select video modes" screen: the video modes which are
  displayed with a 'B' suffix in the number of colors are modes which have
  no custom programming - they use the BIOS and are S-L-O-W ones.

  See "Video Adapter Notes" (p. 123) for comments about particular
  adapters.

  See "Disk-Video" Modes (p. 125) for a description of these non-display
  modes.

  See "Customized Video Modes, FRACTINT.CFG" (p. 126) for information
  about adding your own video modes.


 1.14 Browse Commands

  The following keystrokes function while browsing an image:

  <ARROW KEYS>     Step through the outlines on the screen.
  <ENTER>          Selects the image to display.
  <\>,<h>          Recalls the last image selected.
  <D>              Deletes the selected file.
  <R>              Renames the selected file.
  <s>              Saves the current image with the browser boxes
  displayed.
  <ESC>,<l>        Toggles the browse mode off.
  <Ctrl-b>         Brings up the Browser Parameters (p. 121) screen.

                     Fractint Version xx.xx                     Page 29

  This is a "visual directory", here is how it works...
  When 'L' or 'l' is pressed from a fractal display the current directory
  is searched for any saved files that are deeper zooms of the current
  image and their position shown on screen by a box (or crosshairs if the
  box would be too small). See also Browser Parameters (p. 121) for more
  on how this is done.

  One outline flashes, the selected outline can be changed by using the
  cursor keys.  At the moment the outlines are selected in the order that
  they appear in your directory, so don't worry if the flashing window
  jumps all over the place!
  When enter is pressed, the selected image is loaded. In this mode a
  stack of the last sixteen selected filenames is maintained and the '\'
  or 'h' key pops and loads the last image you were looking at.  Using
  this it is possible to set up sequences of images that allow easy
  exploration of your favorite fractal without having to wait for recalc
  once the level of zoom gets too high, great for demos! (also useful for
  keeping track of just exactly where fract532.gif came from :-) )

  You can also use this facility to tidy up your disk: by typing UPPER
  CASE 'D' when a file is selected the browser will delete the file for
  you, after making sure that you really mean it, you must reply to the
  "are you sure" prompts with an UPPER CASE 'Y' and nothing else,
  otherwise the command is ignored. Just to make absolutely sure you don't
  accidentally wipe out the fruits of many hours of cpu time the default
  setting is to have the browser prompt you twice, you can disable the
  second prompt within the parameters screen, however, if you're feeling
  overconfident :-).

  To complement the Delete function there is a rename function, use the
  UPPER CASE 'R' key for this. You need to enter the FULL new file name,
  no .GIF is implied.

  It is possible to save the current image along with all of the displayed
  boxes indicating subimages by pressing the 's' key.  This exits the
  browse mode to save the image and the boxes become a permanent part of
  the image.  Currently, the screen image ends up with stray dots colored
  after it is saved.

  Esc backs out of image selecting mode.

  To find the next outer image, zoom in using page_up, press
  control_enter, ignore the generating image, and press control_L to start
  browsing.  Whatever is boxed around the center is the next outer image!
  POSSIBLE ERRORS:

  "Sorry..I can't find anything"
  The browser can't locate any files which match the file name mask.  See
  Browser Parameters (p. 121)  This is also displayed if you have less
  than 10K of far memory free when you run Fractint.

  "Sorry....  no more space"
  At the moment the browser can only cope with 450 sub images at one time.
  Any subsequent images are ignored, make sure that the minimum image size
  isn't set too small on the parameters screen.

                     Fractint Version xx.xx                     Page 30

  "Sorry .... out of memory"
  The browser has run out of far memory in which to store the pixels
  covered by the sub image boxes.  Try again with the main image at lower
  resolution, and/or reduce the number of TSRs resident in memory when you
  start Fractint.

  "Sorry.... read only file, can't delete"/ "can't rename"
  The file which you were trying to delete or rename has the read only
  attribute set, you'll need to reset this with your operating system
  before you can get rid of it.



 1.15 RDS Commands

  The following keystrokes function while viewing an RDS image:

  <Enter> or <Space>   -- Toggle calibration bars on and off.
  <Ctrl-s> or <k>      -- Return to RDS Parameters Screen.
  <s>                  -- Save RDS image, then restore original.
  <c>, <+>, <->        -- Color cycle RDS image.
  Other keys           -- Exit RDS mode, restore original image, and pass
                          keystroke on to main menu.

  For more about RDS, see Random Dot Stereograms (RDS) (p. 82)



 1.16 Hints

  Remember, you do NOT have to wait for the program to finish a full
  screen display before entering a command. If you see an interesting spot
  you want to zoom in on while the screen is half-done, don't wait -- do
  it! If you think after seeing the first few lines that another video
  mode would look better, go ahead -- Fractint will shift modes and start
  the redraw at once. When it finishes a display, it beeps and waits for
  your next command.

  In general, the most interesting areas are the "border" areas where the
  colors are changing rapidly. Zoom in on them for the best results. The
  first Mandelbrot-set (default) fractal image has a large, solid-colored
  interior that is the slowest to display; there's nothing to be seen by
  zooming there.

  Plotting time is directly proportional to the number of pixels in a
  screen, and hence increases with the resolution of the video mode.  You
  may want to start in a low-resolution mode for quick progress while
  zooming in, and switch to a higher-resolution mode when things get
  interesting. Or use the solid guessing mode and pre-empt with a zoom
  before it finishes. Plotting time also varies with the maximum iteration
  setting, the fractal type, and your choice of drawing mode.  Solid-
  guessing (the default) is fastest, but it can be wrong: perfectionists
  will want to use dual-pass mode (its first-pass preview is handy if you
  might zoom pre-emptively) or single-pass mode.

                     Fractint Version xx.xx                     Page 31

  When you start systematically exploring, you can save time (and hey,
  every little bit helps -- these "objects" are INFINITE, remember!) by
  <S>aving your last screen in a session to a file, and then going
  straight to it the next time by using the command FRACTINT FRACTxxx (the
  .GIF extension is assumed), or by starting Fractint normally and then
  using the <R> command to reload the saved file. Or you could hit <B> to
  create a parameter file entry with the "recipe" for a given image, and
  next time use the <@> command to re-plot it.


 1.17 Fractint on Unix

  Fractint has been ported to Unix to run under X Windows.  This version
  is called "Xfractint".  Xfractint may be obtained by anonymous ftp to
  sprite.Berkeley.EDU, in the file xfractnnn.shar.Z.

  Xfractint is still under development and is not as reliable as the IBM
  PC version.

  Contact Ken Shirriff (shirriff@eng.sun.com) for information on
  Xfractint.

  Xfractint is a straight port of the IBM PC version.  Thus, it uses the
  IBM user interface.  If you do not have function keys, or Xfractint does
  not accept them from your keyboard, use the following key mappings:

       IBM             Unix
       F1 to F10       Shift-1 to Shift-0
       INSERT          I
       DELETE          D
       PAGE_UP         U
       PAGE_DOWN       N
       LEFT_ARROW      H
       RIGHT_ARROW     L
       UP_ARROW        K
       DOWN_ARROW      J
       HOME            O
       END             E
       CTL_PLUS        }
       CTL_MINUS       {

  Xfractint takes the following options:

  -onroot
  Puts the image on the root window.

  -fast
  Uses a faster drawing technique.

  -disk
  Uses disk video.

  -geometry WxH[{+-X}{+-Y}]
  Changes the geometry of the image window.

                     Fractint Version xx.xx                     Page 32

  -display displayname
  Specifies the X11 display to use.

  -private
  Allocates the entire colormap (i.e. more colors).

  -share
  Shares the current colormap.

  -fixcolors n
  Uses only n colors.

  -slowdisplay
  Prevents xfractint from hanging on the title page with slow displays.

  -simple
  Uses simpler keyboard handling, which makes debugging easier.

  Common problems:

  If you get the message "Couldn't find fractint.hlp", you can
  a) Do "setenv FRACTDIR /foo", replacing /foo with the directory
  containing fractint.hlp.
  b) Run xfractint from the directory containing fractint.hlp, or
  c) Copy fractint.hlp to /usr/local/bin/X11/fractint

  If you get the message "Invalid help signature", the problem is due to
  byteorder.  You are probably using a Sun help file on a Dec machine or
  vice versa.

  If xfractint doesn't accept input, try typing into both the graphics
  window and the text window.  On some systems, only one of these works.

  If you are using Openwindows and can't get xfractint to accept input,
  add to your .Xdefaults file:
  OpenWindows.FocusLenience:      True

  If you cannot view the GIFs that xfractint creates, the problem is that
  xfractint creates GIF89a format and your viewer probably only handles
  GIF87a format.  Run "xfractint gif87a=y" to produce GIF87a format.

  Because many shifted characters are used to simulate IBM keys, you can't
  enter capitalized filenames.

                     Fractint Version xx.xx                     Page 33

 2. Fractal Types

  A list of the fractal types and their mathematics can be found in the
  Summary of Fractal Types (p. 139).  Some notes about how Fractint
  calculates them are in "A Little Code" in "Fractals and the PC" (p. 132)
  .

  Fractint starts by default with the Mandelbrot set. You can change that
  by using the command-line argument "TYPE=" followed by one of the
  fractal type names, or by using the <T> command and selecting the type -
  if parameters are needed, you will be prompted for them.

  In the text that follows, due to the limitations of the ASCII character
  set, "a*b" means "a times b", and "a^b" means "a to the power b".



 2.1 The Mandelbrot Set

  (type=mandel)

  This set is the classic: the only one implemented in many plotting
  programs, and the source of most of the printed fractal images published
  in recent years. Like most of the other types in Fractint, it is simply
  a graph: the x (horizontal) and y (vertical) coordinate axes represent
  ranges of two independent quantities, with various colors used to
  symbolize levels of a third quantity which depends on the first two. So
  far, so good: basic analytic geometry.

  Now things get a bit hairier. The x axis is ordinary, vanilla real
  numbers. The y axis is an imaginary number, i.e. a real number times i,
  where i is the square root of -1. Every point on the plane -- in this
  case, your PC's display screen -- represents a complex number of the
  form:

      x-coordinate + i * y-coordinate

  If your math training stopped before you got to imaginary and complex
  numbers, this is not the place to catch up. Suffice it to say that they
  are just as "real" as the numbers you count fingers with (they're used
  every day by electrical engineers) and they can undergo the same kinds
  of algebraic operations.

  OK, now pick any complex number -- any point on the complex plane -- and
  call it C, a constant. Pick another, this time one which can vary, and
  call it Z. Starting with Z=0 (i.e., at the origin, where the real and
  imaginary axes cross), calculate the value of the expression

      Z^2 + C

  Take the result, make it the new value of the variable Z, and calculate
  again. Take that result, make it Z, and do it again, and so on: in
  mathematical terms, iterate the function Z(n+1) = Z(n)^2 + C. For
  certain values of C, the result "levels off" after a while. For all
  others, it grows without limit. The Mandelbrot set you see at the start
  -- the solid-colored lake (blue by default), the blue circles sprouting

                     Fractint Version xx.xx                     Page 34

  from it, and indeed every point of that color -- is the set of all
  points C for which the magnitude of Z is less than 2 after 150
  iterations (150 is the default setting, changeable via the <X> options
  screen or "maxiter=" parameter).  All the surrounding "contours" of
  other colors represent points for which the magnitude of Z exceeds 2
  after 149 iterations (the contour closest to the M-set itself), 148
  iterations, (the next one out), and so on.

  We actually don't test for the magnitude of Z exceeding 2 - we test the
  magnitude of Z squared against 4 instead because it is easier.  This
  value (FOUR usually) is known as the "bailout" value for the
  calculation, because we stop iterating for the point when it is reached.
  The bailout value can be changed on the <Z> options screen but the
  default is usually best.  See also Bailout Test (p. 81).

  Some features of interest:

  1. Use the <X> options screen to increase the maximum number of
  iterations.  Notice that the boundary of the M-set becomes more and more
  convoluted (the technical terms are "wiggly," "squiggly," and "utterly
  bizarre") as the Z-magnitudes for points that were still within the set
  after 150 iterations turn out to exceed 2 after 200, 500, or 1200. In
  fact, it can be proven that the true boundary is infinitely long: detail
  without limit.

  2. Although there appear to be isolated "islands" of blue, zoom in --
  that is, plot for a smaller range of coordinates to show more detail --
  and you'll see that there are fine "causeways" of blue connecting them
  to the main set. As you zoomed, smaller islands became visible; the same
  is true for them. In fact, there are no isolated points in the M-set: it
  is "connected" in a strict mathematical sense.

  3. The upper and lower halves of the first image are symmetric (a fact
  that Fractint makes use of here and in some other fractal types to speed
  plotting). But notice that the same general features -- lobed discs,
  spirals, starbursts -- tend to repeat themselves (although never
  exactly) at smaller and smaller scales, so that it can be impossible to
  judge by eye the scale of a given image.

  4. In a sense, the contour colors are window-dressing: mathematically,
  it is the properties of the M-set itself that are interesting, and no
  information about it would be lost if all points outside the set were
  assigned the same color. If you're a serious, no-nonsense type, you may
  want to cycle the colors just once to see the kind of silliness that
  other people enjoy, and then never do it again. Go ahead. Just once,
  now. We trust you.


 2.2 Julia Sets

  (type=julia)

  These sets were named for mathematician Gaston Julia, and can be
  generated by a simple change in the iteration process described for the
  Mandelbrot Set (p. 33).  Start with a specified value of C, "C-real + i
  * C-imaginary"; use as the initial value of Z "x-coordinate + i * y-

                     Fractint Version xx.xx                     Page 35

  coordinate"; and repeat the same iteration, Z(n+1) = Z(n)^2 + C.

  There is a Julia set corresponding to every point on the complex plane
  -- an infinite number of Julia sets. But the most visually interesting
  tend to be found for the same C values where the M-set image is busiest,
  i.e.  points just outside the boundary. Go too far inside, and the
  corresponding Julia set is a circle; go too far outside, and it breaks
  up into scattered points. In fact, all Julia sets for C within the M-set
  share the "connected" property of the M-set, and all those for C outside
  lack it.

  Fractint's spacebar toggle lets you "flip" between any view of the M-set
  and the Julia set for the point C at the center of that screen. You can
  then toggle back, or zoom your way into the Julia set for a while and
  then return to the M-set. So if the infinite complexity of the M-set
  palls, remember: each of its infinite points opens up a whole new Julia
  set.

  Historically, the Julia sets came first: it was while looking at the M-
  set as an "index" of all the Julia sets' origins that Mandelbrot noticed
  its properties.

  The relationship between the Mandelbrot (p. 33) set and Julia set can
  hold between other sets as well.  Many of Fractint's types are
  "Mandelbrot/Julia" pairs (sometimes called "M-sets" or "J-sets". All
  these are generated by equations that are of the form z(k+1) =
  f(z(k),c), where the function orbit is the sequence z(0), z(1), ..., and
  the variable c is a complex parameter of the equation. The value c is
  fixed for "Julia" sets and is equal to the first two parameters entered
  with the "params=Creal/Cimag" command. The initial orbit value z(0) is
  the complex number corresponding to the screen pixel. For Mandelbrot
  sets, the parameter c is the complex number corresponding to the screen
  pixel. The value z(0) is c plus a perturbation equal to the values of
  the first two parameters.  See the discussion of Mandellambda Sets
  (p. 39).  This approach may or may not be the "standard" way to create
  "Mandelbrot" sets out of "Julia" sets.

  Some equations have additional parameters.  These values are entered as
  the third or fourth params= value for both Julia and Mandelbrot sets.
  The variables x and y refer to the real and imaginary parts of z;
  similarly, cx and cy are the real and imaginary parts of the parameter c
  and fx(z) and fy(z) are the real and imaginary parts of f(z). The
  variable c is sometimes called lambda for historical reasons.

  NOTE: if you use the "PARAMS=" argument to warp the M-set by starting
  with an initial value of Z other than 0, the M-set/J-sets correspondence
  breaks down and the spacebar toggle no longer works.


 2.3 Julia Toggle Spacebar Commands

  The spacebar toggle has been enhanced for the classic Mandelbrot and
  Julia types. When viewing the Mandelbrot, the spacebar turns on a window
  mode that displays the Inverse Julia corresponding to the cursor
  position in a window.  Pressing the spacebar then causes the regular
  Julia escape time fractal corresponding to the cursor position to be

                     Fractint Version xx.xx                     Page 36

  generated. The following keys take effect in Inverse Julia mode.

  <Space>     Generate the escape-time Julia Set corresponding to the
  cursor
              position. Only works if fractal is a "Mandelbrot" type.
  <n>         Numbers toggle - shows coordinates of the cursor on the
              screen. Press <n> again to turn off numbers.
  <p>         Enter new pixel coordinates directly
  <h>         Hide fractal toggle. Works only if View Windows is turned on
              and set for a small window (such as the default size.) Hides
              the fractal, allowing the orbit to take up the whole screen.
              Press <h> again to uncover the fractal.
  <s>         Saves the fractal, cursor, orbits, and numbers.
  <<> or <,>  Zoom inverse julia image smaller.
  <>> or <.>  Zoom inverse julia image larger.
  <z>         Restore default zoom.

  The Julia Inverse window is only implemented for the classic Mandelbrot
  (type=mandel). For other "Mandelbrot" types <space> turns on the cursor
  without the Julia window, and allows you to select coordinates of the
  matching Julia set in a way similar to the use of the zoom box with the
  Mandelbrot/Julia toggle in previous Fractint versions.


 2.4 Inverse Julias

  (type=julia_inverse)

  Pick a function, such as the familiar Z(n) = Z(n-1) squared plus C (the
  defining function of the Mandelbrot Set).  If you pick a point Z(0) at
  random from the complex plane, and repeatedly apply the function to it,
  you get a sequence of new points called an orbit, which usually either
  zips out toward infinity or zooms in toward one or more "attractor"
  points near the middle of the plane.  The set of all points that are
  "attracted" to infinity is called the "Basin of Attraction" of infinity.
  Each of the other attractors also has its own Basin of Attraction.  Why
  is it called a Basin?  Imagine a lake, and all the water in it
  "draining" into the attractor.  The boundary between these basins is
  called the Julia Set of the function.

  The boundary between the basins of attraction is sort of like a
  repeller; all orbits move away from it, toward one of the attractors.
  But if we define a new function as the inverse of the old one, as for
  instance Z(n) = sqrt(Z(n-1) minus C), then the old attractors become
  repellers, and the former boundary itself becomes the attractor!  Now,
  starting from any point, all orbits are drawn irresistibly to the Julia
  Set!  In fact, once an orbit reaches the boundary, it will continue to
  hop about until it traces the entire Julia Set!  This method for drawing
  Julia Sets is called the Inverse Iteration Method, or IIM for short.

  Unfortunately, some parts of each Julia Set boundary are far more
  attractive to inverse orbits than others are, so that as an orbit traces
  out the set, it keeps coming back to these attractive parts again and
  again, only occasionally visiting the less attractive parts.  Thus it
  may take an infinite length of time to draw the entire set.  To hasten
  the process, we can keep track of how many times each pixel on our

                     Fractint Version xx.xx                     Page 37

  computer screen is visited by an orbit, and whenever an orbit reaches a
  pixel that has already been visited more than a certain number of times,
  we can consider that orbit finished and move on to another one.  This
  "hit limit" thus becomes similar to the iteration limit used in the
  traditional escape-time fractal algorithm.  This is called the Modified
  Inverse Iteration Method, or MIIM, and is much faster than the IIM.

  Now, the inverse of Mandelbrot's classic function is a square root, and
  the square root actually has two solutions; one positive, one negative.
  Therefore at each step of each orbit of the inverse function there is a
  decision; whether to use the positive or the negative square root.  Each
  one gives rise to a new point on the Julia Set, so each is a good
  choice.  This series of choices defines a binary decision tree, each
  point on the Julia Set giving rise to two potential child points.  There
  are many interesting ways to traverse a binary tree, among them Breadth
  first, Depth first (left or negative first), Depth first (right or
  positive first), and completely at random.  It turns out that most
  traversal methods lead to the same or similar pictures, but that how the
  image evolves as the orbits trace it out differs wildly depending on the
  traversal method chosen.  As far as we know, this fact is an original
  discovery by Michael Snyder, and version 18.2 of FRACTINT was its first
  publication.

  Pick a Julia constant such as Z(0) = (-.74543, .11301), the popular
  Seahorse Julia, and try drawing it first Breadth first, then Depth first
  (right first), Depth first (left first), and finally with Random Walk.

  Caveats: the video memory is used in the algorithm, to keep track of how
  many times each pixel has been visited (by changing it's color).
  Therefore the algorithm will not work well if you zoom in far enough
  that part of the Julia Set is off the screen.

  Bugs:   Not working with Disk Video.
          Not resumeable.

  The <J> key toggles between the Inverse Julia orbit and the
  corresponding Julia escape time fractal.


 2.5 Newton domains of attraction

  (type=newtbasin)

  The Newton formula is an algorithm used to find the roots of polynomial
  equations by successive "guesses" that converge on the correct value as
  you feed the results of each approximation back into the formula. It
  works very well -- unless you are unlucky enough to pick a value that is
  on a line BETWEEN two actual roots. In that case, the sequence explodes
  into chaos, with results that diverge more and more wildly as you
  continue the iteration.

  This fractal type shows the results for the polynomial Z^n - 1, which
  has n roots in the complex plane. Use the <T>ype command and enter
  "newtbasin" in response to the prompt. You will be asked for a
  parameter, the "order" of the equation (an integer from 3 through 10 --
  3 for x^3-1, 7 for x^7-1, etc.). A second parameter is a flag to turn on

                     Fractint Version xx.xx                     Page 38

  alternating shades showing changes in the number of iterations needed to
  attract an orbit. Some people like stripes and some don't, as always,
  Fractint gives you a choice!

  The coloring of the plot shows the "basins of attraction" for each root
  of the polynomial -- i.e., an initial guess within any area of a given
  color would lead you to one of the roots. As you can see, things get a
  bit weird along certain radial lines or "spokes," those being the lines
  between actual roots. By "weird," we mean infinitely complex in the good
  old fractal sense. Zoom in and see for yourself.

  This fractal type is symmetric about the origin, with the number of
  "spokes" depending on the order you select. It uses floating-point math
  if you have an FPU, or a somewhat slower integer algorithm if you don't
  have one.


 2.6 Newton

  (type=newton)

  The generating formula here is identical to that for newtbasin (p. 37),
  but the coloring scheme is different. Pixels are colored not according
  to the root that would be "converged on" if you started using Newton's
  formula from that point, but according to the iteration when the value
  is close to a root.  For example, if the calculations for a particular
  pixel converge to the 7th root on the 23rd iteration, NEWTBASIN will
  color that pixel using color #7, but NEWTON will color it using color
  #23.

  If you have a 256-color mode, use it: the effects can be much livelier
  than those you get with type=newtbasin, and color cycling becomes, like,
  downright cosmic. If your "corners" choice is symmetrical, Fractint
  exploits the symmetry for faster display.

  The applicable "params=" values are the same as newtbasin. Try
  "params=4."  Other values are 3 through 10. 8 has twice the symmetry and
  is faster. As with newtbasin, an FPU helps.


 2.7 Complex Newton

  (type=complexnewton/complexbasin)

  Well, hey, "Z^n - 1" is so boring when you can use "Z^a - b" where "a"
  and "b" are complex numbers!  The new "complexnewton" and "complexbasin"
  fractal types are just the old "newton" (p. 38) and "newtbasin"
  (p. 37) fractal types with this little added twist.  When you select
  these fractal types, you are prompted for four values (the real and
  imaginary portions of "a" and "b").  If "a" has a complex portion, the
  fractal has a discontinuity along the negative axis - relax, we finally
  figured out that it's *supposed* to be there!

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 2.8 Lambda Sets

  (type=lambda)

  This type calculates the Julia set of the formula lambda*Z*(1-Z). That
  is, the value Z[0] is initialized with the value corresponding to each
  pixel position, and the formula iterated. The pixel is colored according
  to the iteration when the sum of the squares of the real and imaginary
  parts exceeds 4.

  Two parameters, the real and imaginary parts of lambda, are required.
  Try 0 and 1 to see the classical fractal "dragon". Then try 0.2 and 1
  for a lot more detail to zoom in on.

  It turns out that all quadratic Julia-type sets can be calculated using
  just the formula z^2+c (the "classic" Julia"), so that this type is
  redundant, but we include it for reason of it's prominence in the
  history of fractals.


 2.9 Mandellambda Sets

  (type=mandellambda)

  This type is the "Mandelbrot equivalent" of the lambda (p. 39) set.  A
  comment is in order here. Almost all the Fractint "Mandelbrot" sets are
  created from orbits generated using formulas like z(n+1) = f(z(n),C),
  with z(0) and C initialized to the complex value corresponding to the
  current pixel. Our reasoning was that "Mandelbrots" are maps of the
  corresponding "Julias".  Using this scheme each pixel of a "Mandelbrot"
  is colored the same as the Julia set corresponding to that pixel.
  However, Kevin Allen informs us that the MANDELLAMBDA set appears in the
  literature with z(0) initialized to a critical point (a point where the
  derivative of the formula is zero), which in this case happens to be the
  point (.5,0). Since Kevin knows more about Dr. Mandelbrot than we do,
  and Dr. Mandelbrot knows more about fractals than we do, we defer!
  Starting with version 14 Fractint calculates MANDELAMBDA Dr.
  Mandelbrot's way instead of our way. But ALL THE OTHER "Mandelbrot" sets
  in Fractint are still calculated OUR way!  (Fortunately for us, for the
  classic Mandelbrot Set these two methods are the same!)

  Well now, folks, apart from questions of faithfulness to fractals named
  in the literature (which we DO take seriously!), if a formula makes a
  beautiful fractal, it is not wrong. In fact some of the best fractals in
  Fractint are the results of mistakes! Nevertheless, thanks to Kevin for
  keeping us accurate!

  (See description of "initorbit=" command in Image Calculation Parameters
  (p. 104) for a way to experiment with different orbit intializations).

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 2.10 Circle

  (type=circle)

  This fractal types is from A. K. Dewdney's "Computer Recreations" column
  in "Scientific American". It is attributed to John Connett of the
  University of Minnesota.

  (Don't tell anyone, but this fractal type is not really a fractal!)

  Fascinating Moire patterns can be formed by calculating x^2 + y^2 for
  each pixel in a piece of the complex plane. After multiplication by a
  magnification factor (the parameter), the number is truncated to an
  integer and mapped to a color via color = value modulo (number of
  colors). That is, the integer is divided by the number of colors, and
  the remainder is the color index value used.  The resulting image is not
  a fractal because all detail is lost after zooming in too far. Try it
  with different resolution video modes - the results may surprise you!


 2.11 Plasma Clouds

  (type=plasma)

  Plasma clouds ARE real live fractals, even though we didn't know it at
  first. They are generated by a recursive algorithm that randomly picks
  colors of the corner of a rectangle, and then continues recursively
  quartering previous rectangles. Random colors are averaged with those of
  the outer rectangles so that small neighborhoods do not show much
  change, for a smoothed-out, cloud-like effect. The more colors your
  video mode supports, the better.  The result, believe it or not, is a
  fractal landscape viewed as a contour map, with colors indicating
  constant elevation.  To see this, save and view with the <3> command
  (see "3D" Images (p. 85)) and your "cloud" will be converted to a
  mountain!

  You've GOT to try color cycling (p. 16) on these (hit "+" or "-").  If
  you haven't been hypnotized by the drawing process, the writhing colors
  will do it for sure. We have now implemented subliminal messages to
  exploit the user's vulnerable state; their content varies with your bank
  balance, politics, gender, accessibility to a Fractint programmer, and
  so on. A free copy of Microsoft C to the first person who spots them.

  This type accepts four parameters.

  The first determines how abruptly the colors change. A value of .5
  yields bland clouds, while 50 yields very grainy ones. The default value
  is 2.

  The second determines whether to use the original algorithm (0) or a
  modified one (1). The new one gives the same type of images but draws
  the dots in a different order. It will let you see what the final image
  will look like much sooner than the old one.

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  The third determines whether to use a new seed for generating the next
  plasma cloud (0) or to use the previous seed (1).

  The fourth parameter turns on 16-bit .POT output which provides much
  smoother height gradations. This is especially useful for creating
  mountain landscapes when using the plasma output with a ray tracer such
  as POV-Ray.

  With parameter three set to 1, the next plasma cloud generated will be
  identical to the previous but at whatever new resolution is desired.

  Zooming is ignored, as each plasma-cloud screen is generated randomly.

  The random number seed used for each plasma image is displayed on the
  <tab> information screen, and can be entered with the command line
  parameter "rseed=" to recreate a particular image.

  The algorithm is based on the Pascal program distributed by Bret Mulvey
  as PLASMA.ARC. We have ported it to C and integrated it with Fractint's
  graphics and animation facilities. This implementation does not use
  floating-point math. The algorithm was modified starting with version 18
  so that the plasma effect is independent of screen resolution.

  Saved plasma-cloud screens are EXCELLENT starting images for fractal
  "landscapes" created with the "3D" commands (p. 25).


 2.12 Lambdafn

  (type=lambdafn)

  Function=[sin|cos|sinh|cosh|exp|log|sqr|...]) is specified with this
  type.  Prior to version 14, these types were lambdasine, lambdacos,
  lambdasinh, lambdacos, and lambdaexp.  Where we say "lambdasine" or some
  such below, the good reader knows we mean "lambdafn with function=sin".)

  These types calculate the Julia set of the formula lambda*fn(Z), for
  various values of the function "fn", where lambda and Z are both
  complex.  Two values, the real and imaginary parts of lambda, should be
  given in the "params=" option.  For the feathery, nested spirals of
  LambdaSines and the frost-on-glass patterns of LambdaCosines, make the
  real part = 1, and try values for the imaginary part ranging from 0.1 to
  0.4 (hint: values near 0.4 have the best patterns). In these ranges the
  Julia set "explodes". For the tongues and blobs of LambdaExponents, try
  a real part of 0.379 and an imaginary part of 0.479.

  A coprocessor used to be almost mandatory: each LambdaSine/Cosine
  iteration calculates a hyperbolic sine, hyperbolic cosine, a sine, and a
  cosine (the LambdaExponent iteration "only" requires an exponent, sine,
  and cosine operation)!  However, Fractint now computes these
  transcendental functions with fast integer math. In a few cases the fast
  math is less accurate, so we have kept the old slow floating point code.
  To use the old code, invoke with the float=yes option, and, if you DON'T
  have a coprocessor, go on a LONG vacation!

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 2.13 Mandelfn

  (type=mandelfn)

  Function=[sin|cos|sinh|cosh|exp|log|sqr|...]) is specified with this
  type.  Prior to version 14, these types were mandelsine, mandelcos,
  mandelsinh, mandelcos, and mandelexp. Same comment about our lapses into
  the old terminology as above!

  These are "pseudo-Mandelbrot" mappings for the LambdaFn (p. 41) Julia
  functions.  They map to their corresponding Julia sets via the spacebar
  command in exactly the same fashion as the original M/J sets.  In
  general, they are interesting mainly because of that property (the
  function=exp set in particular is rather boring). Generate the
  appropriate "Mandelfn" set, zoom on a likely spot where the colors are
  changing rapidly, and hit the spacebar key to plot the Julia set for
  that particular point.

  Try "FRACTINT TYPE=MANDELFN CORNERS=4.68/4.76/-.03/.03 FUNCTION=COS" for
  a graphic demonstration that we're not taking Mandelbrot's name in vain
  here. We didn't even know these little buggers were here until Mark
  Peterson found this a few hours before the version incorporating
  Mandelfns was released.

  Note: If you created images using the lambda or mandel "fn" types prior
  to version 14, and you wish to update the fractal information in the
  "*.fra" file, simply read the files and save again. You can do this in
  batch mode via a command line such as:

       "fractint oldfile.fra savename=newfile.gif batch=yes"

  For example, this procedure can convert a version 13 "type=lambdasine"
  image to a version 14 "type=lambdafn function=sin" GIF89a image.  We do
  not promise to keep this "backward compatibility" past version 14 - if
  you want to keep the fractal information in your *.fra files accurate,
  we recommend conversion.  See GIF Save File Format (p. 166).


 2.14 Barnsley Mandelbrot/Julia Sets

  (type=barnsleym1/.../j3)

  Michael Barnsley has written a fascinating college-level text, "Fractals
  Everywhere," on fractal geometry and its graphic applications. (See
  Bibliography (p. 169).) In it, he applies the principle of the M and J
  sets to more general functions of two complex variables.

  We have incorporated three of Barnsley's examples in Fractint. Their
  appearance suggests polarized-light microphotographs of minerals, with
  patterns that are less organic and more crystalline than those of the
  M/J sets. Each example has both a "Mandelbrot" and a "Julia" type.
  Toggle between them using the spacebar.

  The parameters have the same meaning as they do for the "regular"
  Mandelbrot and Julia. For types M1, M2, and M3, they are used to "warp"
  the image by setting the initial value of Z. For the types J1 through

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  J3, they are the values of C in the generating formulas.

  Be sure to try the <O>rbit function while plotting these types.


 2.15 Barnsley IFS Fractals

  (type=ifs)

  One of the most remarkable spin-offs of fractal geometry is the ability
  to "encode" realistic images in very small sets of numbers -- parameters
  for a set of functions that map a region of two-dimensional space onto
  itself.  In principle (and increasingly in practice), a scene of any
  level of complexity and detail can be stored as a handful of numbers,
  achieving amazing "compression" ratios... how about a super-VGA image of
  a forest, more than 300,000 pixels at eight bits apiece, from a 1-KB
  "seed" file?

  Again, Michael Barnsley and his co-workers at the Georgia Institute of
  Technology are to be thanked for pushing the development of these
  iterated function systems (IFS).

  When you select this fractal type, Fractint scans the current IFS file
  (default is FRACTINT.IFS, a set of definitions supplied with Fractint)
  for IFS definitions, then prompts you for the IFS name you wish to run.
  Fern and 3dfern are good ones to start with. You can press <F6> at the
  selection screen if you want to select a different .IFS file you've
  written.

  Note that some Barnsley IFS values generate images quite a bit smaller
  than the initial (default) screen. Just bring up the zoom box, center it
  on the small image, and hit <Enter> to get a full-screen image.

  To change the number of dots Fractint generates for an IFS image before
  stopping, you can change the "maximum iterations" parameter on the <X>
  options screen.

  Fractint supports two types of IFS images: 2D and 3D. In order to fully
  appreciate 3D IFS images, since your monitor is presumably 2D, we have
  added rotation, translation, and perspective capabilities. These share
  values with the same variables used in Fractint's other 3D facilities;
  for their meaning see "Rectangular Coordinate Transformation" (p. 90).
  You can enter these values from the command line using:

  rotation=xrot/yrot/zrot       (try 30/30/30)
  shift=xshift/yshift           (shifts BEFORE applying perspective!)
  perspective=viewerposition    (try 200)

  Alternatively, entering <I> from main screen will allow you to modify
  these values. The defaults are the same as for regular 3D, and are not
  always optimum for 3D IFS. With the 3dfern IFS type, try
  rotation=30/30/30. Note that applying shift when using perspective
  changes the picture -- your "point of view" is moved.

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  A truly wild variation of 3D may be seen by entering "2" for the stereo
  mode (see "Stereo 3D Viewing" (p. 89)), putting on red/blue "funny
  glasses", and watching the fern develop with full depth perception right
  there before your eyes!

  This feature USED to be dedicated to Bruce Goren, as a bribe to get him
  to send us MORE knockout stereo slides of 3D ferns, now that we have
  made it so easy! Bruce, what have you done for us *LATELY* ?? (Just
  kidding, really!)

  Each line in an IFS definition (look at FRACTINT.IFS with your editor
  for examples) contains the parameters for one of the generating
  functions, e.g. in FERN:
     a    b     c    d    e    f    p
   ___________________________________
     0     0    0  .16    0    0   .01
   .85   .04 -.04  .85    0  1.6   .85
   .2   -.26  .23  .22    0  1.6   .07
  -.15   .28  .26  .24    0  .44   .07

  The values on each line define a matrix, vector, and probability:
      matrix   vector  prob
      |a b|     |e|     p
      |c d|     |f|

  The "p" values are the probabilities assigned to each function (how
  often it is used), which add up to one. Fractint supports up to 32
  functions, although usually three or four are enough.

  3D IFS definitions are a bit different.  The name is followed by (3D) in
  the definition file, and each line of the definition contains 13
  numbers: a b c d e f g h i j k l p, defining:
      matrix   vector  prob
      |a b c|   |j|     p
      |d e f|   |k|
      |g h i|   |l|

  The program FDESIGN can be used to design IFS fractals - see FDESIGN
  (p. 171).

  You can save the points in your IFS fractal in the file ORBITS.RAW which
  is overwritten each time a fractal is generated. The program Acrospin
  can read this file and will let you view the fractal from any angle
  using the cursor keys. See Acrospin (p. 171).


 2.16 Sierpinski Gasket

  (type=sierpinski)

  Another pre-Mandelbrot classic, this one found by W. Sierpinski around
  World War I. It is generated by dividing a triangle into four congruent
  smaller triangles, doing the same to each of them, and so on, yea, even
  unto infinity. (Notice how hard we try to avoid reiterating
  "iterating"?)

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  If you think of the interior triangles as "holes", they occupy more and
  more of the total area, while the "solid" portion becomes as hopelessly
  fragile as that gasket you HAD to remove without damaging it -- you
  remember, that Sunday afternoon when all the parts stores were closed?
  There's a three-dimensional equivalent using nested tetrahedrons instead
  of triangles, but it generates too much pyramid power to be safely
  unleashed yet.

  There are no parameters for this type. We were able to implement it with
  integer math routines, so it runs fairly quickly even without an FPU.


 2.17 Quartic Mandelbrot/Julia

  (type=mandel4/julia4)

  These fractal types are the moral equivalent of the original M and J
  sets, except that they use the formula Z(n+1) = Z(n)^4 + C, which adds
  additional pseudo-symmetries to the plots. The "Mandel4" set maps to the
  "Julia4" set via -- surprise! -- the spacebar toggle. The M4 set is kind
  of boring at first (the area between the "inside" and the "outside" of
  the set is pretty thin, and it tends to take a few zooms to get to any
  interesting sections), but it looks nice once you get there. The Julia
  sets look nice right from the start.

  Other powers, like Z(n)^3 or Z(n)^7, work in exactly the same fashion.
  We used this one only because we're lazy, and Z(n)^4 = (Z(n)^2)^2.


 2.18 Distance Estimator

  (distest=nnn/nnn)

  This used to be type=demm and type=demj.  These types have not died, but
  are only hiding!  They are equivalent to the mandel and julia types with
  the "distest=" option selected with a predetermined value.

  The Distance Estimator Method (p. 74) can be used to produce higher
  quality images of M and J sets, especially suitable for printing in
  black and white.

  If you have some *.fra files made with the old types demm/demj, you may
  want to convert them to the new form.  See the Mandelfn (p. 42) section
  for directions to carry out the conversion.


 2.19 Pickover Mandelbrot/Julia Types

  (type=manfn+zsqrd/julfn+zsqrd, manzpowr/julzpowr, manzzpwr/julzzpwr,
  manfn+exp/julfn+exp - formerly included man/julsinzsqrd and
  man/julsinexp which have now been generalized)

  These types have been explored by Clifford A. Pickover, of the IBM
  Thomas J. Watson Research center. As implemented in Fractint, they are
  regular Mandelbrot/Julia set pairs that may be plotted with or without
  the "biomorph" (p. 78) option Pickover used to create organic-looking

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  beasties (see below). These types are produced with formulas built from
  the functions z^z, z^n, sin(z), and e^z for complex z. Types with
  "power" or "pwr" in their name have an exponent value as a third
  parameter. For example, type=manzpower params=0/0/2 is our old friend
  the classical Mandelbrot, and type=manzpower params=0/0/4 is the Quartic
  Mandelbrot. Other values of the exponent give still other fractals.
  Since these WERE the original "biomorph" types, we should give an
  example.  Try:

      FRACTINT type=manfn+zsqrd biomorph=0 corners=-8/8/-6/6 function=sin

  to see a big biomorph digesting little biomorphs!


 2.20 Pickover Popcorn

  (type=popcorn/popcornjul)

  Here is another Pickover idea. This one computes and plots the orbits of
  the dynamic system defined by:

           x(n+1) = x(n) - h*sin(y(n)+tan(3*y(n))
           y(n+1) = y(n) - h*sin(x(n)+tan(3*x(n))

  with the initializers x(0) and y(0) equal to ALL the complex values
  within the "corners" values, and h=.01.  ALL these orbits are
  superimposed, resulting in "popcorn" effect.  You may want to use a
  maxiter value less than normal - Pickover recommends a value of 50.  As
  a bonus, type=popcornjul shows the Julia set generated by these same
  equations with the usual escape-time coloring. Turn on orbit viewing
  with the "O" command, and as you watch the orbit pattern you may get
  some insight as to where the popcorn comes from. Although you can zoom
  and rotate popcorn, the results may not be what you'd expect, due to the
  superimposing of orbits and arbitrary use of color. Just for fun we
  added type popcornjul, which is the plain old Julia set calculated from
  the same formula.


 2.21 Peterson Variations

  (type=marksmandel, marksjulia, cmplxmarksmand, cmplxmarksjul,
  marksmandelpwr, tim's_error)

  These fractal types are contributions of Mark Peterson. MarksMandel and
  MarksJulia are two families of fractal types that are linked in the same
  manner as the classic Mandelbrot/Julia sets: each MarksMandel set can be
  considered as a mapping into the MarksJulia sets, and is linked with the
  spacebar toggle. The basic equation for these sets is:
        Z(n+1) = ((lambda^exp-1) * Z(n)^2) + lambda where Z(0) = 0.0 and
  lambda is (x + iy) for MarksMandel. For MarksJulia, Z(0) = (x + iy) and
  lambda is a constant (taken from the MarksMandel spacebar toggle, if
  that method is used). The exponent is a positive integer or a complex
  number. We call these "families" because each value of the exponent
  yields a different MarksMandel set, which turns out to be a kinda-
  polygon with (exponent) sides. The exponent value is the third
  parameter, after the "initialization warping" values. Typically one

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  would use null warping values, and specify the exponent with something
  like "PARAMS=0/0/5", which creates an unwarped, pentagonal MarksMandel
  set.

  In the process of coding MarksMandelPwr formula type, Tim Wegner created
  the type "tim's_error" after making an interesting coding mistake.


 2.22 Unity

  (type=unity)

  This Peterson variation began with curiosity about other "Newton-style"
  approximation processes. A simple one,

     One = (x * x) + (y * y); y = (2 - One) * x;   x = (2 - One) * y;

  produces the fractal called Unity.

  One of its interesting features is the "ghost lines." The iteration loop
  bails out when it reaches the number 1 to within the resolution of a
  screen pixel. When you zoom a section of the image, the bailout
  criterion is adjusted, causing some lines to become thinner and others
  thicker.

  Only one line in Unity that forms a perfect circle: the one at a radius
  of 1 from the origin. This line is actually infinitely thin. Zooming on
  it reveals only a thinner line, up (down?) to the limit of accuracy for
  the algorithm. The same thing happens with other lines in the fractal,
  such as those around |x| = |y| = (1/2)^(1/2) = .7071

  Try some other tortuous approximations using the TEST stub (p. 54) and
  let us know what you come up with!


 2.23 Scott Taylor / Lee Skinner Variations

  (type=fn(z*z), fn*fn, fn*z+z, fn+fn, fn+fn(pix), sqr(1/fn), sqr(fn),
  spider, tetrate, manowar)

  Two of Fractint's faithful users went bonkers when we introduced the
  "formula" type, and came up with all kinds of variations on escape-time
  fractals using trig functions.  We decided to put them in as regular
  types, but there were just too many! So we defined the types with
  variable functions and let you, the overwhelmed user, specify what the
  functions should be! Thus Scott Taylor's "z = sin(z) + z^2" formula type
  is now the "fn+fn" regular type, and EITHER function can be one of sin,
  cos, tan, cotan, sinh, cosh, tanh, cotanh, exp, log, sqr, recip, ident,
  conj, flip, cosxx, asin, asinh, acos, acosh, atan, atanh, sqrt, abs, or
  cabs.

  Plus we give you 4 parameters to set, the complex coefficients of the
  two functions!  Thus the innocent-looking "fn+fn" type is really 256
  different types in disguise, not counting the damage done by the
  parameters!

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   Some functions that require further explanation:

   conj()   - returns the complex conjugate of the argument. That is,
   changes
              sign of the imaginary component of argument: (x,y) becomes (x,-y)
   ident()  - identity function. Leaves the value of the argument
   unchanged,
              acting like a "z" term in a formula.
   flip()   - Swap the real and imaginary components of the complex
   number.
              e.g. (4,5) would become (5,4)

  Lee informs us that you should not judge fractals by their "outer"
  appearance. For example, the images produced by z = sin(z) + z^2 and z =
  sin(z) - z^2 look very similar, but are different when you zoom in.


 2.24 Kam Torus

  (type=kamtorus, kamtorus3d)

  This type is created by superimposing orbits generated by a set of
  equations, with a variable incremented each time.

           x(0) = y(0) = orbit/3;
           x(n+1) = x(n)*cos(a) + (x(n)*x(n)-y(n))*sin(a)
           y(n+1) = x(n)*sin(a) - (x(n)*x(n)-y(n))*cos(a)

  After each orbit, 'orbit' is incremented by a step size. The parameters
  are angle "a", step size for incrementing 'orbit', stop value for
  'orbit', and points per orbit. Try this with a stop value of 5 with
  sound=x for some weird fractal music (ok, ok, fractal noise)! You will
  also see the KAM Torus head into some chaotic territory that Scott
  Taylor wanted to hide from you by setting the defaults the way he did,
  but now we have revealed all!

  The 3D variant is created by treating 'orbit' as the z coordinate.

  With both variants, you can adjust the "maxiter" value (<X> options
  screen or parameter maxiter=) to change the number of orbits plotted.


 2.25 Bifurcation

  (type=bifxxx)

  The wonder of fractal geometry is that such complex forms can arise from
  such simple generating processes. A parallel surprise has emerged in the
  study of dynamical systems: that simple, deterministic equations can
  yield chaotic behavior, in which the system never settles down to a
  steady state or even a periodic loop. Often such systems behave normally
  up to a certain level of some controlling parameter, then go through a
  transition in which there are two possible solutions, then four, and
  finally a chaotic array of possibilities.

                     Fractint Version xx.xx                     Page 49

  This emerged many years ago in biological models of population growth.
  Consider a (highly over-simplified) model in which the rate of growth is
  partly a function of the size of the current population:

  New Population =  Growth Rate * Old Population * (1 - Old Population)

  where population is normalized to be between 0 and 1. At growth rates
  less than 200 percent, this model is stable: for any starting value,
  after several generations the population settles down to a stable level.
  But for rates over 200 percent, the equation's curve splits or
  "bifurcates" into two discrete solutions, then four, and soon becomes
  chaotic.

  Type=bifurcation illustrates this model. (Although it's now considered a
  poor one for real populations, it helped get people thinking about
  chaotic systems.) The horizontal axis represents growth rates, from 190
  percent (far left) to 400 percent; the vertical axis normalized
  population values, from 0 to 4/3. Notice that within the chaotic region,
  there are narrow bands where there is a small, odd number of stable
  values. It turns out that the geometry of this branching is fractal;
  zoom in where changing pixel colors look suspicious, and see for
  yourself.

  Three parameters apply to bifurcations: Filter Cycles, Seed Population,
  and Function or Beta.

  Filter Cycles (default 1000) is the number of iterations to be done
  before plotting maxiter population values. This gives the iteration time
  to settle into the characteristic patterns that constitute the
  bifurcation diagram, and results in a clean-looking plot.  However,
  using lower values produces interesting results too. Set Filter Cycles
  to 1 for an unfiltered map.

  Seed Population (default 0.66) is the initial population value from
  which all others are calculated. For filtered maps the final image is
  independent of Seed Population value in the valid range (0.0 < Seed
  Population < 1.0).
  Seed Population becomes effective in unfiltered maps - try setting
  Filter Cycles to 1 (unfiltered) and Seed Population to 0.001
  ("PARAMS=1/.001" on the command line). This results in a map overlaid
  with nice curves. Each Seed Population value results in a different set
  of curves.

  Function (default "ident") is the function applied to the old population
  before the new population is determined. The "ident" function calculates
  the same bifurcation fractal that was generated before these formulae
  were generalized.

  Beta is used in the bifmay bifurcations and is the power to which the
  denominator is raised.

  Note that fractint normally uses periodicity checking to speed up
  bifurcation computation.  However, in some cases a better quality image
  will be obtained if you turn off periodicity checking with
  "periodicity=no"; for instance, if you use a high number of iterations
  and a smooth colormap.

                     Fractint Version xx.xx                     Page 50

  Many formulae can be used to produce bifurcations.  Mitchel Feigenbaum
  studied lots of bifurcations in the mid-70's, using a HP-65 calculator
  (IBM PCs, Fractals, and Fractint, were all Sci-Fi then !). He studied
  where bifurcations occurred, for the formula r*p*(1-p), the one
  described above.  He found that the ratios of lengths of adjacent areas
  of bifurcation were four and a bit.  These ratios vary, but, as the
  growth rate increases, they tend to a limit of 4.669+.  This helped him
  guess where bifurcation points would be, and saved lots of time.

  When he studied bifurcations of r*sin(PI*p) he found a similar pattern,
  which is not surprising in itself.  However, 4.669+ popped out, again.
  Different formulae, same number ?  Now, THAT's surprising !  He tried
  many other formulae and ALWAYS got 4.669+ - Hot Damn !!!  So hot, in
  fact, that he phoned home and told his Mom it would make him Famous ! He
  also went on to tell other scientists.  The rest is History...

  (It has been conjectured that if Feigenbaum had a copy of Fractint, and
  used it to study bifurcations, he may never have found his Number, as it
  only became obvious from long perusal of hand-written lists of values,
  without the distraction of wild color-cycling effects !).
  We now know that this number is as universal as PI or E. It appears in
  situations ranging from fluid-flow turbulence, electronic oscillators,
  chemical reactions, and even the Mandelbrot Set - yup, fraid so:
  "budding" of the Mandelbrot Set along the negative real axis occurs at
  intervals determined by Feigenbaum's Number, 4.669201660910.....

  Fractint does not make direct use of the Feigenbaum Number (YET !).
  However, it does now reflect the fact that there is a whole sub-species
  of Bifurcation-type fractals.  Those implemented to date, and the
  related formulae, (writing P for pop[n+1] and p for pop[n]) are :

    bifurcation  P =  p + r*fn(p)*(1-fn(p))  Verhulst Bifurcations.
    biflambda    P =      r*fn(p)*(1-fn(p))  Real equivalent of Lambda
    Sets.
    bif+sinpi    P =  p + r*fn(PI*p)         Population scenario based
    on...
    bif=sinpi    P =      r*fn(PI*p)         ...Feigenbaum's second
    formula.
    bifstewart   P =      r*fn(p)*fn(p) - 1  Stewart Map.
    bifmay       P =      r*p / ((1+p)^b)    May Map.

  It took a while for bifurcations to appear here, despite them being over
  a century old, and intimately related to chaotic systems. However, they
  are now truly alive and well in Fractint!


 2.26 Orbit Fractals

  Orbit Fractals are generated by plotting an orbit path in two or three
  dimensional space.

  See Lorenz Attractors (p. 51), Rossler Attractors (p. 52), Henon
  Attractors (p. 52), Pickover Attractors (p. 53), Gingerbreadman
  (p. 53), and Martin Attractors (p. 53).

                     Fractint Version xx.xx                     Page 51

  The orbit trajectory for these types can be saved in the file ORBITS.RAW
  by invoking Fractint with the "orbitsave=yes" command-line option.  This
  file will be overwritten each time you generate a new fractal, so rename
  it if you want to save it.  A nifty program called Acrospin can read
  these files and rapidly rotate them in 3-D - see Acrospin (p. 171).


 2.27 Lorenz Attractors

  (type=lorenz/lorenz3d)

  The "Lorenz Attractor" is a "simple" set of three deterministic
  equations developed by Edward Lorenz while studying the non-
  repeatability of weather patterns.  The weather forecaster's basic
  problem is that even very tiny changes in initial patterns ("the beating
  of a butterfly's wings" - the official term is "sensitive dependence on
  initial conditions") eventually reduces the best weather forecast to
  rubble.

  The lorenz attractor is the plot of the orbit of a dynamic system
  consisting of three first order non-linear differential equations. The
  solution to the differential equation is vector-valued function of one
  variable.  If you think of the variable as time, the solution traces an
  orbit.  The orbit is made up of two spirals at an angle to each other in
  three dimensions. We change the orbit color as time goes on to add a
  little dazzle to the image.  The equations are:

                  dx/dt = -a*x + a*y
                  dy/dt =  b*x - y   -z*x
                  dz/dt = -c*z + x*y

  We solve these differential equations approximately using a method known
  as the first order taylor series.  Calculus teachers everywhere will
  kill us for saying this, but you treat the notation for the derivative
  dx/dt as though it really is a fraction, with "dx" the small change in x
  that happens when the time changes "dt".  So multiply through the above
  equations by dt, and you will have the change in the orbit for a small
  time step. We add these changes to the old vector to get the new vector
  after one step. This gives us:

               xnew = x + (-a*x*dt) + (a*y*dt)
               ynew = y + (b*x*dt) - (y*dt) - (z*x*dt)
               znew = z + (-c*z*dt) + (x*y*dt)

               (default values: dt = .02, a = 5, b = 15, c = 1)

  We connect the successive points with a line, project the resulting 3D
  orbit onto the screen, and voila! The Lorenz Attractor!

  We have added two versions of the Lorenz Attractor.  "Type=lorenz" is
  the Lorenz attractor as seen in everyday 2D.  "Type=lorenz3d" is the
  same set of equations with the added twist that the results are run
  through our perspective 3D routines, so that you get to view it from
  different angles (you can modify your perspective "on the fly" by using
  the <I> command.)  If you set the "stereo" option to "2", and have
  red/blue funny glasses on, you will see the attractor orbit with depth

                     Fractint Version xx.xx                     Page 52

  perception.

  Hint: the default perspective values (x = 60, y = 30, z = 0) aren't the
  best ones to use for fun Lorenz Attractor viewing.  Experiment a bit -
  start with rotation values of 0/0/0 and then change to 20/0/0 and 40/0/0
  to see the attractor from different angles.- and while you're at it, use
  a non-zero perspective point Try 100 and see what happens when you get
  *inside* the Lorenz orbits.  Here comes one - Duck!  While you are at
  it, turn on the sound with the "X". This way you'll at least hear it
  coming!

  Different Lorenz attractors can be created using different parameters.
  Four parameters are used. The first is the time-step (dt). The default
  value is .02. A smaller value makes the plotting go slower; a larger
  value is faster but rougher. A line is drawn to connect successive orbit
  values.  The 2nd, third, and fourth parameters are coefficients used in
  the differential equation (a, b, and c). The default values are 5, 15,
  and 1.  Try changing these a little at a time to see the result.


 2.28 Rossler Attractors

  (type=rossler3D)

  This fractal is named after the German Otto Rossler, a non-practicing
  medical doctor who approached chaos with a bemusedly philosophical
  attitude.  He would see strange attractors as philosophical objects. His
  fractal namesake looks like a band of ribbon with a fold in it. All we
  can say is we used the same calculus-teacher-defeating trick of
  multiplying the equations by "dt" to solve the differential equation and
  generate the orbit.  This time we will skip straight to the orbit
  generator - if you followed what we did above with type Lorenz (p. 51)
  you can easily reverse engineer the differential equations.

               xnew = x - y*dt -   z*dt
               ynew = y + x*dt + a*y*dt
               znew = z + b*dt + x*z*dt - c*z*dt

  Default parameters are dt = .04, a = .2, b = .2, c = 5.7


 2.29 Henon Attractors

  (type=henon)

  Michel Henon was an astronomer at Nice observatory in southern France.
  He came to the subject of fractals via investigations of the orbits of
  astronomical objects.  The strange attractor most often linked with
  Henon's name comes not from a differential equation, but from the world
  of discrete mathematics - difference equations. The Henon map is an
  example of a very simple dynamic system that exhibits strange behavior.
  The orbit traces out a characteristic banana shape, but on close
  inspection, the shape is made up of thicker and thinner parts.  Upon
  magnification, the thicker bands resolve to still other thick and thin
  components.  And so it goes forever! The equations that generate this
  strange pattern perform the mathematical equivalent of repeated

                     Fractint Version xx.xx                     Page 53

  stretching and folding, over and over again.

               xnew =  1 + y - a*x*x
               ynew =  b*x

  The default parameters are a=1.4 and b=.3.


 2.30 Pickover Attractors

  (type=pickover)

  Clifford A. Pickover of the IBM Thomas J. Watson Research center is such
  a creative source for fractals that we attach his name to this one only
  with great trepidation.  Probably tomorrow he'll come up with another
  one and we'll be back to square one trying to figure out a name!

  This one is the three dimensional orbit defined by:

               xnew = sin(a*y) - z*cos(b*x)
               ynew = z*sin(c*x) - cos(d*y)
               znew = sin(x)

  Default parameters are: a = 2.24, b = .43, c = -.65, d = -2.43


 2.31 Gingerbreadman

  (type=gingerbreadman)

  This simple fractal is a charming example stolen from "Science of
  Fractal Images", p. 149.

               xnew = 1 - y + |x|
               ynew = x

  The initial x and y values are set by parameters, defaults x=-.1, y = 0.


 2.32 Martin Attractors

  (type=hopalong/martin)

  These fractal types are from A. K. Dewdney's "Computer Recreations"
  column in "Scientific American". They are attributed to Barry Martin of
  Aston University in Birmingham, England.

  Hopalong is an "orbit" type fractal like lorenz. The image is obtained
  by iterating this formula after setting z(0) = y(0) = 0:
        x(n+1) = y(n) - sign(x(n))*sqrt(abs(b*x(n)-c))
        y(n+1) = a - x(n)
  Parameters are a, b, and c. The function "sign()"  returns 1 if the
  argument is positive, -1 if argument is negative.

                     Fractint Version xx.xx                     Page 54

  This fractal continues to develop in surprising ways after many
  iterations.

  Another Martin fractal is simpler. The iterated formula is:
        x(n+1) = y(n) - sin(x(n))
        y(n+1) = a - x(n)
  The parameter is "a". Try values near the number pi.

  Michael Peters has based the HOP program on variations of these Martin
  types.  You will find three of these here: chip, quadruptwo, and
  threeply.


 2.33 Icon

  (type=icon/icon3d)

    This fractal type was inspired by the book "Symmetry in Chaos" by
    Michael Field and Martin Golubitsky (ISBN 0-19-853689-5, Oxford Press)

    To quote from the book's jacket,

      "Field and Golubitsky describe how a chaotic process eventually can
      lead to symmetric patterns (in a river, for instance, photographs of
      the turbulent movement of eddies, taken over time, often reveal
      patterns on the average."

    The Icon type implemented here maps the classic population logistic
    map of bifurcation fractals onto the complex plane in Dn symmetry.

    The initial points plotted are the more chaotic initial orbits, but as
    you wait, delicate webs will begin to form as the orbits settle into a
    more periodic pattern.  Since pixels are colored by the number of
    times they are hit, the more periodic paths will become clarified with
    time.  These fractals run continuously.

  There are 6 parameters:  Lambda, Alpha, Beta, Gamma, Omega, and Degree
      Omega  0 = Dn, or dihedral (rotation + reflectional) symmetry
            !0 = Zn, or cyclic (rotational) symmetry
      Degree = n, or Degree of symmetry


 2.34 Test

  (type=test)

  This is a stub that we (and you!) use for trying out new fractal types.
  "Type=test" fractals make use of Fractint's structure and features for
  whatever code is in the routine 'testpt()' (located in the small source
  file TESTPT.C) to determine the color of a particular pixel.

  If you have a favorite fractal type that you believe would fit nicely
  into Fractint, just rewrite the C function in TESTPT.C (or use the
  prototype function there, which is a simple M-set implementation) with
  an algorithm that computes a color based on a point in the complex
  plane.

                     Fractint Version xx.xx                     Page 55

  After you get it working, send your code to one of the authors and we
  might just add it to the next release of Fractint, with full credit to
  you. Our criteria are: 1) an interesting image and 2) a formula
  significantly different from types already supported. (Bribery may also
  work. THIS author is completely honest, but I don't trust those other
  guys.) Be sure to include an explanation of your algorithm and the
  parameters supported, preferably formatted as you see here to simplify
  folding it into the documentation.


 2.35 Formula

  (type=formula)

  This is a "roll-your-own" fractal interpreter - you don't even need a
  compiler!

  To run a "type=formula" fractal, you first need a text file containing
  formulas (there's a sample file - FRACTINT.FRM - included with this
  distribution).  When you select the "formula" fractal type, Fractint
  scans the current formula file (default is FRACTINT.FRM) for formulas,
  then prompts you for the formula name you wish to run.  After prompting
  for any parameters, the formula is parsed for syntax errors and then the
  fractal is generated. If you want to use a different formula file, press
  <F6> when you are prompted to select a formula name.

  There are two command-line options that work with type=formula
  ("formulafile=" and "formulaname="), useful when you are using this
  fractal type in batch mode.

  The following documentation is supplied by Mark Peterson, who wrote the
  formula interpreter:

  Formula fractals allow you to create your own fractal formulas.  The
  general format is:

     Mandelbrot(XAXIS) { z = Pixel:  z = sqr(z) + pixel, |z| <= 4 }
        |         |          |                |              |
       Name     Symmetry    Initial         Iteration       Bailout
                            Condition                       Criteria

  Initial conditions are set, then the iterations performed while the
  bailout criteria remains true or until 'z' turns into a periodic loop.
  All variables are created automatically by their usage and treated as
  complex.  If you declare 'v = 2' then the variable 'v' is treated as a
  complex with an imaginary value of zero.

            Predefined Variables (x, y)
            --------------------------------------------
            z          used for periodicity checking
            p1         parameters 1 and 2
            p2         parameters 3 and 4
            p3         parameters 5 and 6
            pixel      screen coordinates
            LastSqr    Modulus from the last sqr() function
            rand       Complex random number

                     Fractint Version xx.xx                     Page 56

            Precedence
            --------------------------------------------
            1          sin(), cos(), sinh(), cosh(), cosxx(), tan(), cotan(),
                       tanh(), cotanh(), sqr(), log(), exp(), abs(), conj(),
                       real(), imag(), flip(), fn1(), fn2(), fn3(), fn4(),
                       srand(), asin(), asinh(), acos(), acosh(), atan(),
                       atanh(), sqrt(), cabs()
            2          - (negation), ^ (power)
            3          * (multiplication), / (division)
            4          + (addition), - (subtraction)
            5          = (assignment)
            6          < (less than), <= (less than or equal to)
                       > (greater than), >= (greater than or equal to)
                       == (equal to), != (not equal to)
            7          && (logical AND), || (logical OR)

  Precedence may be overridden by use of parenthesis.  Note the modulus
  squared operator |z| is also parenthetic and always sets the imaginary
  component to zero.  This means 'c * |z - 4|' first subtracts 4 from z,
  calculates the modulus squared then multiplies times 'c'.  Nested
  modulus squared operators require overriding parenthesis: c * |z +
  (|pixel|)|

  The functions fn1(...) to fn4(...) are variable functions - when used,
  the user is prompted at run time (on the <Z> screen) to specify one of
  sin, cos, sinh, cosh, exp, log, sqr, etc. for each required variable
  function.

  Most of the functions have their conventional meaning, here are a few
  notes on others that are not conventional. The function cosxx()
  duplicates a bug in the version 16 cos() function. Then abs(x+iy) =
  abs(x)+i*abs(y), flip(x+iy) = y+i*x, and |x+iy| = x*x+y*y.

  The formulas are performed using either integer or floating point
  mathematics depending on the <F> floating point toggle.  If you do not
  have an FPU then type MPC math is performed in lieu of traditional
  floating point.

  The 'rand' predefined variable is changed with each iteration to a new
  random number with the real and imaginary components containing a value
  between zero and 1. Use the srand() function to initialize the random
  numbers to a consistent random number sequence.  If a formula does not
  contain the srand() function, then the formula compiler will use the
  system time to initialize the sequence.  This could cause a different
  fractal to be generated each time the formula is used depending on how
  the formula is written.

  Remember that when using integer math there is a limited dynamic range,
  so what you think may be a fractal could really be just a limitation of
  the integer math range.  God may work with integers, but His dynamic
  range is many orders of magnitude greater than our puny 32 bit
  mathematics!  Always verify with the floating point <F> toggle.

  The possible values for symmetry are:

                     Fractint Version xx.xx                     Page 57

  XAXIS,  XAXIS_NOPARM
  YAXIS,  YAXIS_NOPARM
  XYAXIS, XYAXIS_NOPARM
  ORIGIN, ORIGIN_NOPARM
  PI_SYM, PI_SYM_NOPARM
  XAXIS_NOREAL
  XAXIS_NOIMAG

  These will force the symmetry even if no symmetry is actually present,
  so try your formulas without symmetry before you use these.

  For mathematical formulas of functions used in the parser language, see
   Trig Identities (p. 154)


 2.36 Julibrots

  (type=julibrot)

  The Julibrot fractal type uses a general-purpose renderer for
  visualizing three dimensional solid fractals. Originally Mark Peterson
  developed this rendering mechanism to view a 3-D sections of a 4-D
  structure he called a "Julibrot".  This structure, also called "layered
  Julia set" in the fractal literature, hinges on the relationship between
  the Mandelbrot and Julia sets. Each Julia set is created using a fixed
  value c in the iterated formula z^2 + c. The Julibrot is created by
  layering Julia sets in the x-y plane and continuously varying c,
  creating new Julia sets as z is incremented. The solid shape thus
  created is rendered by shading the surface using a brightness inversely
  proportional to the virtual viewer's eye.

  Starting with Fractint version 18, the Julibrot engine can be used with
  other Julia formulas besides the classic z^2 + c. The first field on the
  Julibrot parameter screen lets you select which orbit formula to use.

  You can also use the Julibrot renderer to visualize 3D cross sections of
  true four dimensional Quaternion and Hypercomplex fractals.

  The Julibrot Parameter Screens

  Orbit Algorithm - select the orbit algorithm to use. The available
     possibilities include 2-D Julia and both mandelbrot and Julia
     variants of the 4-D Quaternion and Hypercomplex fractals.

  Orbit parameters - the next screen lets you fill in any parameters
     belonging to the orbit algorithm. This list of parameters is not
     necessarily the same as the list normally presented for the orbit
     algorithm, because some of these parameters are used in the Julibrot
     layering process.

     From/To Parameters These parameters allow you to specify the
     "Mandelbrot" values used to generate the layered Julias. The
     parameter c in the Julia formulas will be incremented in steps
     ranging from the "from" x and y values to the "to" x and y values. If
     the orbit formula is one of the "true" four dimensional fractal types
     quat, quatj, hypercomplex, or hypercomplexj, then these numbers are

                     Fractint Version xx.xx                     Page 58

     used with the 3rd and 4th dimensional values.

     The "from/to" variables are different for the different kinds of
     orbit algorithm.

        2D Julia sets - complex number formula z' = f(z) + c
           The "from/to" parameters change the values of c.
        4D Julia sets - Quaternion or Hypercomplex formula z' = f(z) + c
           The four dimensions of c are set by the orbit parameters.
           The first two dimensions of z are determined by the corners values.
           The third and fourth dimensions of z are the "to/from" variables.
        4D Mandelbrot sets - Quaternion or Hypercomplex formula z' = f(z)
        + c
           The first two dimensions of c are determined by the corners values.
           The third and fourth dimensions of c are the "to/from" variables.

  Distance between the eyes - set this to 2.5 if you want a red/blue
     anaglyph image, 0 for a normal greyscale image.

  Number of z pixels - this sets how many layers are rendered in the
     screen z-axis. Use a higher value with higher resolution video modes.

  The remainder of the parameters are needed to construct the red/blue
  picture so that the fractal appears with the desired depth and proper
  'z' location.  With the origin set to 8 inches beyond the screen plane
  and the depth of the fractal at 8 inches the default fractal will appear
  to start at 4 inches beyond the screen and extend to 12 inches if your
  eyeballs are 2.5 inches apart and located at a distance of 24 inches
  from the screen.  The screen dimensions provide the reference frame.



 2.37 Diffusion Limited Aggregation

  (type=diffusion)

  This type begins with a single point in the center of the screen.
  Subsequent points move around randomly until coming into contact with
  the first point, at which time their locations are fixed and they are
  colored randomly.  This process repeats until the fractals reaches the
  edge of the screen.  Use the show orbits function to see the points'
  random motion.

  One unfortunate problem is that on a large screen, this process will
  tend to take eons.  To speed things up, the points are restricted to a
  box around the initial point.  The first and only parameter to diffusion
  contains the size of the border between the fractal and the edge of the
  box.  If you make this number small, the fractal will look more solid
  and will be generated more quickly.

  Diffusion was inspired by a Scientific American article a couple of
  years back which includes actual pictures of real physical phenomena
  that behave like this.

                     Fractint Version xx.xx                     Page 59

  Thanks to Adrian Mariano for providing the diffusion code and
  documentation. Juan J. Buhler added the additional options.


 2.38 Magnetic Fractals

  (type=magnet1m/.../magnet2j)

  These fractals use formulae derived from the study of hierarchical
  lattices, in the context of magnetic renormalisation transformations.
  This kinda stuff is useful in an area of theoretical physics that deals
  with magnetic phase-transitions (predicting at which temperatures a
  given substance will be magnetic, or non-magnetic).  In an attempt to
  clarify the results obtained for Real temperatures (the kind that you
  and I can feel), the study moved into the realm of Complex Numbers,
  aiming to spot Real phase-transitions by finding the intersections of
  lines representing Complex phase-transitions with the Real Axis.  The
  first people to try this were two physicists called Yang and Lee, who
  found the situation a bit more complex than first expected, as the phase
  boundaries for Complex temperatures are (surprise!) fractals.

  And that's all the technical (?) background you're getting here!  For
  more details (are you SERIOUS ?!) read "The Beauty of Fractals".  When
  you understand it all, you might like to rewrite this section, before
  you start your new job as a professor of theoretical physics...

  In Fractint terms, the important bits of the above are "Fractals",
  "Complex Numbers", "Formulae", and "The Beauty of Fractals".  Lifting
  the Formulae straight out of the Book and iterating them over the
  Complex plane (just like the Mandelbrot set) produces Fractals.

  The formulae are a bit more complicated than the Z^2+C used for the
  Mandelbrot Set, that's all.  They are :

                    [               ] 2
                    |  Z^2 + (C-1)  |
          MAGNET1 : | ------------- |
                    |  2*Z + (C-2)  |
                    [               ]

                    [                                         ] 2
                    |      Z^3 + 3*(C-1)*Z + (C-1)*(C-2)      |
          MAGNET2 : | --------------------------------------- |
                    |  3*(Z^2) + 3*(C-2)*Z + (C-1)*(C-2) + 1  |
                    [                                         ]

  These aren't quite as horrific as they look (oh yeah ?!) as they only
  involve two variables (Z and C), but cubing things, doing division, and
  eventually squaring the result (all in Complex Numbers) don't exactly
  spell S-p-e-e-d !  These are NOT the fastest fractals in Fractint !

  As you might expect, for both formulae there is a single related
  Mandelbrot-type set (magnet1m, magnet2m) and an infinite number of
  related Julia-type sets (magnet1j, magnet2j), with the usual toggle
  between the corresponding Ms and Js via the spacebar.

                     Fractint Version xx.xx                     Page 60

  If you fancy delving into the Julia-types by hand, you will be prompted
  for the Real and Imaginary parts of the parameter denoted by C.  The
  result is symmetrical about the Real axis (and therefore the initial
  image gets drawn in half the usual time) if you specify a value of Zero
  for the Imaginary part of C.

  Fractint Historical Note:  Another complication (besides the formulae)
  in implementing these fractal types was that they all have a finite
  attractor (1.0 + 0.0i), as well as the usual one (Infinity).  This fact
  spurred the development of Finite Attractor logic in Fractint.  Without
  this code you can still generate these fractals, but you usually end up
  with a pretty boring image that is mostly deep blue "lake", courtesy of
  Fractint's standard Periodicity Logic (p. 134).  See Finite Attractors
  (p. 152) for more information on this aspect of Fractint internals.

  (Thanks to Kevin Allen for Magnetic type documentation above).


 2.39 L-Systems

  (type=lsystem)

  These fractals are constructed from line segments using rules specified
  in drawing commands.  Starting with an initial string, the axiom,
  transformation rules are applied a specified number of times, to produce
  the final command string which is used to draw the image.

  Like the type=formula fractals, this type requires a separate data file.
  A sample file, FRACTINT.L, is included with this distribution.  When you
  select type lsystem, the current lsystem file is read and you are asked
  for the lsystem name you wish to run. Press <F6> at this point if you
  wish to use a different lsystem file. After selecting an lsystem, you
  are asked for one parameter - the "order", or number of times to execute
  all the transformation rules.  It is wise to start with small orders,
  because the size of the substituted command string grows exponentially
  and it is very easy to exceed your resolution.  (Higher orders take
  longer to generate too.)  The command line options "lname=" and "lfile="
  can be used to over-ride the default file name and lsystem name.

  Each L-System entry in the file contains a specification of the angle,
  the axiom, and the transformation rules.  Each item must appear on its
  own line and each line must be less than 160 characters long.

  The statement "angle n" sets the angle to 360/n degrees; n must be an
  integer greater than two and less than fifty.

  "Axiom string" defines the axiom.

  Transformation rules are specified as "a=string" and convert the single
  character 'a' into "string."  If more than one rule is specified for a
  single character all of the strings will be added together.  This allows
  specifying transformations longer than the 160 character limit.
  Transformation rules may operate on any characters except space, tab or
  '}'.

                     Fractint Version xx.xx                     Page 61

  Any information after a ; (semi-colon) on a line is treated as a
  comment.

  Here is a sample lsystem:

  Dragon {         ; Name of lsystem, { indicates start
    Angle 8        ; Specify the angle increment to 45 degrees
    Axiom FX       ; Starting character string
    F=             ; First rule:  Delete 'F'
    y=+FX--FY+     ; Change 'y' into  "+fx--fy+"
    x=-FX++FY-     ; Similar transformation on 'x'
  }                ; final } indicates end

  The standard drawing commands are:
      F Draw forward
      G Move forward (without drawing)
      + Increase angle
      - Decrease angle
      | Try to turn 180 degrees. (If angle is odd, the turn
        will be the largest possible turn less than 180 degrees.)

  These commands increment angle by the user specified angle value. They
  should be used when possible because they are fast. If greater
  flexibility is needed, use the following commands which keep a
  completely separate angle pointer which is specified in degrees.

      D   Draw forward
      M   Move forward
      \nn Increase angle nn degrees
      /nn Decrease angle nn degrees

  Color control:
      Cnn Select color nn
      <nn Increment color by nn
      >nn decrement color by nn

  Advanced commands:
      !     Reverse directions (Switch meanings of +, - and , /)
      @nnn  Multiply line segment size by nnn
            nnn may be a plain number, or may be preceded by
                I for inverse, or Q for square root.
                (e.g.  @IQ2 divides size by the square root of 2)
      [     Push.  Stores current angle and position on a stack
      ]     Pop.  Return to location of last push

  Other characters are perfectly legal in command strings.  They are
  ignored for drawing purposes, but can be used to achieve complex
  translations.

  The characters '+', '-', '<', '>', '[', ']', '|', '!', '@', '/', '\',
  and 'c' are reserved symbols and cannot be redefined.  For example,
  c=f+f and <= , are syntax errors.

  The integer code produces incorrect results in five known instances,
  Peano2 with order >= 7, SnowFlake1 with order >=6, and SnowFlake2,
  SnowFlake3, and SnowflakeColor with order >= 5.  If you see strange

                     Fractint Version xx.xx                     Page 62

  results, switch to the floating point code.


 2.40 Lyapunov Fractals

  (type=lyapunov)

  The Bifurcation fractal illustrates what happens in a simple population
  model as the growth rate increases.  The Lyapunov fractal expands that
  model into two dimensions by letting the growth rate vary in a periodic
  fashion between two values.  Each pair of growth rates is run through a
  logistic population model and a value called the Lyapunov Exponent is
  calculated for each pair and is plotted. The Lyapunov Exponent is
  calculated by adding up log | r - 2*r*x| over many cycles of the
  population model and dividing by the number of cycles. Negative Lyapunov
  exponents indicate a stable, periodic behavior and are plotted in color.
  Positive Lyapunov exponents indicate chaos (or a diverging model) and
  are colored black.

  Order parameter.  Each possible periodic sequence yields a two
  dimensional space to explore.  The Order parameter selects a sequence.
  The default value 0 represents the sequence ab which alternates between
  the two values of the growth parameter.  On the screen, the a values run
  vertically and the b values run horizontally. Here is how to calculate
  the space parameter for any desired sequence.  Take your sequence of a's
  and b's and arrange it so that it starts with at least 2 a's and ends
  with a b. It may be necessary to rotate the sequence or swap a's and
  b's. Strike the first a and the last b off the list and replace each
  remaining a with a 1 and each remaining b with a zero.  Interpret this
  as a binary number and convert it into decimal.

  An Example.  I like sonnets.  A sonnet is a poem with fourteen lines
  that has the following rhyming sequence: abba abba abab cc.  Ignoring
  the rhyming couplet at the end, let's calculate the Order parameter for
  this pattern.

    abbaabbaabab         doesn't start with at least 2 a's
    aabbaabababb         rotate it
    1001101010           drop the first and last, replace with 0's and 1's

    512+64+32+8+2 = 618

  An Order parameter of 618 gives the Lyapunov equivalent of a sonnet.
  "How do I make thee? Let me count the ways..."

  Population Seed.  When two parts of a Lyapunov overlap, which spike
  overlaps which is strongly dependent on the initial value of the
  population model.  Any changes from using a different starting value
  between 0 and 1 may be subtle. The values 0 and 1 are interpreted in a
  special manner. A Seed of 1 will choose a random number between 0 and 1
  at the start of each pixel. A Seed of 0 will suppress resetting the seed
  value between pixels unless the population model diverges in which case
  a random seed will be used on the next pixel.

                     Fractint Version xx.xx                     Page 63

  Filter Cycles.  Like the Bifurcation model, the Lyapunov allow you to
  set the number of cycles that will be run to allow the model to approach
  equilibrium before the lyapunov exponent calculation is begun. The
  default value of 0 uses one half of the iterations before beginning the
  calculation of the exponent.

  Reference.  A.K. Dewdney, Mathematical Recreations, Scientific American,
  Sept. 1991


 2.41 fn||fn Fractals

  (type=lambda(fn||fn), manlam(fn||fn), julia(fn||fn), mandel(fn||fn))

  Two functions=[sin|cos|sinh|cosh|exp|log|sqr|...]) are specified with
  these types.  The two functions are alternately used in the calculation
  based on a comparison between the modulus of the current Z and the shift
  value.  The first function is used if the modulus of Z is less than the
  shift value and the second function is used otherwise.

  The lambda(fn||fn) type calculates the Julia set of the formula
  lambda*fn(Z), for various values of the function "fn", where lambda and
  Z are both complex.  Two values, the real and imaginary parts of lambda,
  should be given in the "params=" option.  The third value is the shift
  value.  The space bar will generate the corresponding "pseudo
  Mandelbrot" set, manlam(fn||fn).

  The manlam(fn||fn) type calculates the "pseudo Mandelbrot" set of the
  formula fn(Z)*C, for various values of the function "fn", where C and Z
  are both complex.  Two values, the real and imaginary parts of Z(0),
  should be given in the "params=" option.  The third value is the shift
  value.  The space bar will generate the corresponding julia set,
  lamda(fn||fn).

  The julia(fn||fn) type calculates the Julia set of the formula fn(Z)+C,
  for various values of the function "fn", where C and Z are both complex.
  Two values, the real and imaginary parts of C, should be given in the
  "params=" option.  The third value is the shift value.  The space bar
  will generate the corresponding mandelbrot set, mandel(fn||fn).

  The mandel(fn||fn) type calculates the Mandelbrot set of the formula
  fn(Z)+C, for various values of the function "fn", where C and Z are both
  complex.  Two values, the real and imaginary parts of Z(0), should be
  given in the "params=" option.  The third value is the shift value.  The
  space bar will generate the corresponding julia set, julia(fn||fn).


 2.42 Halley

  (type=halley)

  The Halley map is an algorithm used to find the roots of polynomial
  equations by successive "guesses" that converge on the correct value as
  you feed the results of each approximation back into the formula. It
  works very well -- unless you are unlucky enough to pick a value that is
  on a line BETWEEN two actual roots. In that case, the sequence explodes

                     Fractint Version xx.xx                     Page 64

  into chaos, with results that diverge more and more wildly as you
  continue the iteration.

  This fractal type shows the results for the polynomial Z(Z^a - 1), which
  has a+1 roots in the complex plane. Use the <T>ype command and enter
  "halley" in response to the prompt. You will be asked for a parameter,
  the "order" of the equation (an integer from 2 through 10 -- 2 for Z(Z^2
  - 1), 7 for Z(Z^7 - 1), etc.). A second parameter is the relaxation
  coefficient, and is used to control the convergence stability. A number
  greater than one increases the chaotic behavior and a number less than
  one decreases the chaotic behavior. The third parameter is the value
  used to determine when the formula has converged. The test for
  convergence is ||Z(n+1)|^2 - |Z(n)|^2| < epsilon. This convergence test
  produces the whisker-like projections which generally point to a root.


 2.43 Dynamic System

  (type=dynamic, dynamic2)

  These fractals are based on a cyclic system of differential equations:
           x'(t) = -f(y(t))
           y'(t) = f(x(t))
  These equations are approximated by using a small time step dt, forming
  a time-discrete dynamic system:
           x(n+1) = x(n) - dt*f(y(n))
           y(n+1) = y(n) + dt*f(x(n))
  The initial values x(0) and y(0) are set to various points in the plane,
  the dynamic system is iterated, and the resulting orbit points are
  plotted.

  In fractint, the function f is restricted to: f(k) = sin(k + a*fn1(b*k))
  The parameters are the spacing of the initial points, the time step dt,
  and the parameters (a,b,fn1) that affect the function f.  Normally the
  orbit points are plotted individually, but for a negative spacing the
  points are connected.

  This fractal is similar to the Pickover Popcorn (p. 46).
  A variant is the implicit Euler approximation:
           y(n+1) = y(n) + dt*f(x(n))
           x(n+1) = x(n) - dt*f(y(n+1))
  This variant results in complex orbits.  The implicit Euler
  approximation is selected by entering dt<0.

  There are two options that have unusual effects on these fractals.  The
  Orbit Delay value controls how many initial points are computed before
  the orbits are displayed on the screen.  This allows the orbit to settle
  down.  The outside=summ option causes each pixel to increment color
  every time an orbit touches it; the resulting display is a 2-d
  histogram.

  These fractals are discussed in Chapter 14 of Pickover's "Computers,
  Pattern, Chaos, and Beauty".

                     Fractint Version xx.xx                     Page 65

 2.44 Mandelcloud

  (type=mandelcloud)

  This fractal computes the Mandelbrot function, but displays it
  differently.  It starts with regularly spaced initial pixels and
  displays the resulting orbits.  This idea is somewhat similar to the
  Dynamic System (p. 64).

  There are two options that have unusual effects on this fractal.  The
  Orbit Delay value controls how many initial points are computed before
  the orbits are displayed on the screen.  This allows the orbit to settle
  down.  The outside=summ option causes each pixel to increment color
  every time an orbit touches it; the resulting display is a 2-d
  histogram.

  This fractal was invented by Noel Giffin.



 2.45 Quaternion

  (type=quat,quatjul)

  These fractals are based on quaternions.  Quaternions are an extension
  of complex numbers, with 4 parts instead of 2.  That is, a quaternion Q
  equals a+ib+jc+kd, where a,b,c,d are reals.  Quaternions have rules for
  addition and multiplication.  The normal Mandelbrot and Julia formulas
  can be generalized to use quaternions instead of complex numbers.

  There is one complication.  Complex numbers have 2 parts, so they can be
  displayed on a plane.  Quaternions have 4 parts, so they require 4
  dimensions to view.  That is, the quaternion Mandelbrot set is actually
  a 4-dimensional object.  Each quaternion C generates a 4-dimensional
  Julia set.

  One method of displaying the 4-dimensional object is to take a 3-
  dimensional slice and render the resulting object in 3-dimensional
  perspective.  Fractint isn't that sophisticated, so it merely displays a
  2-dimensional slice of the resulting object. (Note: Now Fractint is that
  sophisticated!  See the Julibrot type!)

  In fractint, for the Julia set, you can specify the four parameters of
  the quaternion constant: c=(c1,ci,cj,ck), but the 2-dimensional slice of
  the z-plane Julia set is fixed to (xpixel,ypixel,0,0).

  For the Mandelbrot set, you can specify the position of the c-plane
  slice: (xpixel,ypixel,cj,ck).

  These fractals are discussed in Chapter 10 of Pickover's "Computers,
  Pattern, Chaos, and Beauty".

  See also HyperComplex (p. 66) and  Quaternion and Hypercomplex Algebra
  (p. 155)

                     Fractint Version xx.xx                     Page 66

 2.46 HyperComplex

  (type=hypercomplex,hypercomplexj)

  These fractals are based on hypercomplex numbers, which like quaternions
  are a four dimensional generalization of complex numbers. It is not
  possible to fully generalize the complex numbers to four dimensions
  without sacrificing some of the algebraic properties shared by real and
  complex numbers. Quaternions violate the commutative law of
  multiplication, which says z1*z2 = z2*z1. Hypercomplex numbers fail the
  rule that says all non-zero elements have multiplicative inverses - that
  is, if z is not 0, there should be a number 1/z such that (1/z)*(z) = 1.
  This law holds most of the time but not all the time for hypercomplex
  numbers.

  However hypercomplex numbers have a wonderful property for fractal
  purposes.  Every function defined for complex numbers has a simple
  generalization to hypercomplex numbers. Fractint's implementation takes
  advantage of this by using "fn" variables - the iteration formula is
      h(n+1) = fn(h(n)) + C.
  where "fn" is the hypercomplex generalization of sin, cos, log, sqr etc.
  You can see 3D versions of these fractals using fractal type Julibrot.
  Hypercomplex numbers were brought to our attention by Clyde Davenport,
  author of "A Hypercomplex Calculus with Applications to Relativity",
  ISBN 0-9623837-0-8.

  See also Quaternion (p. 65) and  Quaternion and Hypercomplex Algebra
  (p. 155)




 2.47 Cellular Automata

  (type=cellular)

  These fractals are generated by 1-dimensional cellular automata.
  Consider a 1-dimensional line of cells, where each cell can have the
  value 0 or 1.  In each time step, the new value of a cell is computed
  from the old value of the cell and the values of its neighbors.  On the
  screen, each horizontal row shows the value of the cells at any one
  time.  The time axis proceeds down the screen, with each row computed
  from the row above.

  Different classes of cellular automata can be described by how many
  different states a cell can have (k), and how many neighbors on each
  side are examined (r).  Fractint implements the binary nearest neighbor
  cellular automata (k=2,r=1), the binary next-nearest neighbor cellular
  automata (k=2,r=2), and the ternary nearest neighbor cellular automata
  (k=3,r=1) and several others.

  The rules used here determine the next state of a given cell by using
  the sum of the states in the cell's neighborhood.  The sum of the cells
  in the neighborhood are mapped by rule to the new value of the cell.
  For the binary nearest neighbor cellular automata, only the closest
  neighbor on each side is used.  This results in a 4 digit rule

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  controlling the generation of each new line:  if each of the cells in
  the neighborhood is 1, the maximum sum is 1+1+1 = 3 and the sum can
  range from 0 to 3, or 4 values.  This results in a 4 digit rule.  For
  instance, in the rule 1010, starting from the right we have 0->0, 1->1,
  2->0, 3->1.  If the cell's neighborhood sums to 2, the new cell value
  would be 0.

  For the next-nearest cellular automata (kr = 22), each pixel is
  determined from the pixel value and the two neighbors on each side.
  This results in a 6 digit rule.

  For the ternary nearest neighbor cellular automata (kr = 31), each cell
  can have the value 0, 1, or 2.  A single neighbor on each side is
  examined, resulting in a 7 digit rule.

    kr  #'s in rule  example rule     | kr  #'s in rule  example rule
    21      4        1010             | 42     16        2300331230331001
    31      7        1211001          | 23      8        10011001
    41     10        3311100320       | 33     15        021110101210010
    51     13        2114220444030    | 24     10        0101001110
    61     16        3452355321541340 | 25     12        110101011001
    22      6        011010           | 26     14        00001100000110
    32     11        21212002010      | 27     16        0010000000000110

  The starting row of cells can be set to a pattern of up to 16 digits or
  to a random pattern.  The borders are set to zeros if a pattern is
  entered or are set randomly if the starting row is set randomly.

  A zero rule will randomly generate the rule to use.

  Hitting the space bar toggles between continuously generating the
  cellular automata and stopping at the end of the current screen.

  Recommended reading: "Computer Software in Science and Mathematics",
  Stephen Wolfram, Scientific American, September, 1984.  "Abstract
  Mathematical Art", Kenneth E. Perry, BYTE, December, 1986.  "The
  Armchair Universe", A. K. Dewdney, W. H. Freeman and Company, 1988.
  "Complex Patterns Generated by Next Nearest Neighbors Cellular
  Automata", Wentian Li, Computers & Graphics, Volume 13, Number 4.


 2.48 Ant Automaton

  (type=ant)

  This fractal type is the generalized Ant Automaton described in the
  "Computer Recreations" column of the July 1994 Scientific American. The
  article attributes this automaton to Greg Turk of Stanford University,
  Leonid A.  Bunivomitch of the Georgia Institute of Technology, and S. E.
  Troubetzkoy of the University of Bielefeld.

  The ant wanders around the screen, starting at the middle. A rule
  string, which the user can input as Fractint's first parameter,
  determines the ant's direction. This rule string is stored as a double
  precision number in our implementation. Only the digit 1 is significant
  -- all other digits are treated as 0. When the type 1 ant leaves a cell

                     Fractint Version xx.xx                     Page 68

  (a pixel on the screen) of color k, it turns right if the kth symbol in
  the rule string is a 1, or left otherwise. Then the color in the
  abandoned cell is incremented. The type 2 ant uses only the rule string
  to move around. If the digit of the rule string is a 1, the ant turns
  right and puts a zero in current cell, otherwise it turns left and put a
  number in the current cell. An empty rule string causes the rule to be
  generated randomly.

  Fractint's 2nd parameter is a maximum iteration to guarantee that the
  fractal will terminate.

  The 3rd parameter is the number of ants (up to 256). If you select 0
  ants, then the number oif ants is random.

  The 4th paramter allows you to select ant type 1 (the original), or type
  2.

  The 5th parameter determines whether the ant's progress stops when the
  edge of the screen is reaches (as in the original implementation), or
  whether the ant's path wraps to the opposite side of the screen. You can
  slow down the ant to see her better using the <x> screen Orbit Delay -
  try 10.

  The 6th parameter accepts a random seed, allowing you to duplicate
  images using random values (empty rule string or 0 maximum ants.

  Try rule string 10. In this case, the ant moves in a seemingly random
  pattern, then suddenly marches off in a straight line. This happens for
  many other rule strings. The default 1100 produces symmetrical images.

  If the screen initially contains an image, the path of the ant changes.
  To try this, generate a fractal, and press <Ctrl-a>. Note that images
  seeded with an image are not (yet) reproducible in PAR files. When
  started using the <Ctrl-a> keys, after the ant is finished the default
  fractal type reverts to that of the underlying fractal.

  Special keystrokes are in effect during the ant's march. The <space> key
  toggles a step-by-step mode. When in this mode, press <enter> to see
  each step of the ant's progress. When orbit delay (on <x> screen) is set
  to 1, the step mode is the default.

  If you press the right or left arrow during the ant's journey, you can
  adjust the orbit delay factor with the arrow keys (increment by 10) or
  ctrl-arrow keys (increment by 100). Press any other key to get out of
  the orbit delay adjustment mode. Higher values cause slower motion.
  Changed values are not saved after the ant is finished, but you can set
  the orbit delay value in advance from the <x> screen.


 2.49 Phoenix

  (type=phoenix, mandphoenix, phoenixcplx, mandphoenixclx)

  The phoenix type defaults to the original phoenix curve discovered by
  Shigehiro Ushiki, "Phoenix", IEEE Transactions on Circuits and Systems,
  Vol. 35, No. 7, July 1988, pp. 788-789.  These images do not have the X

                     Fractint Version xx.xx                     Page 69

  and Y axis swapped as is normal for this type.

  The mandphoenix type is the corresponding Mandelbrot set image of the
  phoenix type.  The spacebar toggles between the two as long as the
  mandphoenix type has an initial z(0) of (0,0).  The mandphoenix is not
  an effective index to the phoenix type, so explore the wild blue yonder.

  To reproduce the Mandelbrot set image of the phoenix type as shown in
  Stevens' book, "Fractal Programming in C", set initorbit=0/0 on the
  command line or with the <g> key.  The colors need to be rotated one
  position because Stevens uses the values from the previous calculation
  instead of the current calculation to determine when to bailout.

  The phoenixcplx type is implemented using complex constants instead of
  the real constants that Stevens used.  This recreates the mapping as
  originally presented by Ushiki.

  The mandphoenixclx type is the corresponding Mandelbrot set image of the
  phoenixcplx type.  The spacebar toggles between the two as long as the
  mandphoenixclx type has a perturbation of z(0) = (0,0).  The
  mandphoenixclx is an effective index to the phoenixcplx type.


 2.50 Frothy Basins

  (type=frothybasin)

  Frothy Basins, or Riddled Basins, were discovered by James C. Alexander
  of the University of Maryland.  The discussion below is derived from a
  two page article entitled "Basins of Froth" in Science News, November
  14, 1992 and from correspondence with others, including Dr. Alexander.

  The equations that generate this fractal are not very different from
  those that generate many other orbit fractals.

        Z(0) = pixel;
        Z(n+1) = Z(n)^2 - C*conj(Z(n))
        where C = 1 + A*i

  One of the things that makes this fractal so interesting is the shape of
  the dynamical system's attractors.  It is not at all uncommon for a
  dynamical system to have non-point attractors.  Shapes such as circles
  are very common.  Strange attractors are attractors which are themselves
  fractal.  What is unusual about this system, however, is that the
  attractors intersect.  This is the first case in which such a phenomenon
  has been observed.  The attractors for this system are made up of line
  segments which overlap to form an equilateral triangle.  This attractor
  triangle can be seen by using the "show orbits" option (the 'o' key) or
  the "orbits window" option (the ctrl-'o' key).

  The number of attractors present is dependant on the value of A, the
  imaginary part of C.  For values where A <= 1.028713768218725..., there
  are three attractors.  When A is larger than this critical value, two of
  attractors merge into one, leaving only two attractors.  An interesting
  variation on this fractal can be generated by applying the above mapping
  twice per each iteration.  The result is that some of the attractors are

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  split into two parts, giving the system either six or three attractors,
  depending on whether A is less than or greater than the critical value.

  These are also called "Riddled Basins" because each basin is riddled
  with holes.  Which attractor a point is eventually pulled into is
  extremely sensitive to its initial position.  A very slight change in
  any direction may cause it to end up on a different attractor.  As a
  result, the basins are thoroughly intermingled. The effect appears to be
  a frothy mixture that has been subjected to lots of stirring and
  folding.

  Pixel color is determined by which attractor captures the orbit.  The
  shade of color is determined by the number of iterations required to
  capture the orbit.  In Fractint, the actual shade of color used depends
  on how many colors are available in the video mode being used.  If 256
  colors are available, the default coloring scheme is determined by the
  number of iterations that were required to capture the orbit.  An
  alternative coloring scheme can be used where the shade is determined by
  the iterations required divided by the maximum iterations.  This method
  is especially useful on deeply zoomed images.  If only 16 colors are
  available, then only the alternative coloring scheme is used.  If fewer
  than 16 colors are available, then Fractint just colors the basins
  without any shading.

                     Fractint Version xx.xx                     Page 71

 3. Doodads, Bells, and Whistles


 3.1 Drawing Method

  The "passes option" (<X> options screen or "passes=" parameter) selects
  one of the single-pass, dual-pass, triple-pass, solid-guessing
  (default), boundary tracing, or tesseral modes.  This option applies to
  most fractal types.

  Single-pass mode ("1") draws the screen pixel by pixel.

  Dual-pass ("2") generates a half-resolution screen first as a preview
  using 2x2-pixel boxes, and then generates the rest of the dots with a
  second pass. Dual-pass uses no more time than single-pass.

  Triple-pass ("3") generates the coarse first pass of the solidguessing
  mode (see "g" below), then switches to either "1" (with low resolution
  video modes) or "2" (with higher resolution video modes). The advantage
  of '3' vs '2' is that when using high resolution modes, the first pass
  has a much lower resolution (about 160x120) and is therefore much
  quicker than the first pass of the passes=2 mode. However, with the '2'
  mode, the first pass does not represent wasted time. The '3' mode wastes
  the effort of generating the coarse first screen.

  The single, dual, and triple pass modes all result in identical images.
  These modes are for those who desire the highest possible accuracy. Most
  people will want to use the guessing mode, described next.

  Solid-guessing ("g") is the default.  It performs from two to four
  visible passes - more in higher resolution video modes. Its first
  visible pass is actually two passes - one pixel per 4x4, 8x8, or 16x16
  pixel box is generated, and the guessing logic is applied to fill in the
  blocks at the next level (2x2, 4x4, or 8x8).  Subsequent passes fill in
  the display at the next finer resolution, skipping blocks which are
  surrounded by the same color. Solid-guessing can guess wrong, but it
  sure guesses quickly!

  Boundary Tracing ("b"), which only works accurately with fractal types
  (such as the Mandelbrot set, but not the Newton type) that do not
  contain "islands" of colors, finds a color boundary, traces it around
  the screen, and then "blits" in the color over the enclosed area.

  Tesseral ("t") is a sort of "super-solid-guessing" option that
  successively divides the image into subsections and colors in rectangles
  that have a boundary of a solid color. It's actually slower than the
  solid-guessing algorithm, but it looks neat, so we left it in. This mode
  is also subject to errors when islands of color appear inside the
  rectangles.

  The "fillcolor=" option in the <X> screen or on the command line sets a
  fixed color to be used by the Boundary Tracing and Tesseral calculations
  for filling in defined regions. The effect of this is to show off the
  boundaries of the areas delimited by these two methods.

                     Fractint Version xx.xx                     Page 72

 3.2 Palette Maps

  If you have a VGA, MCGA, Super-VGA, 8514/A, XGA, TARGA, or TARGA+ video
  adapter, you can save and restore color palettes for use with any image.
  To load a palette onto an existing image, use the <L> command in color-
  cycling or palette-editing mode.  To save a palette, use the <S> command
  in those modes.  To change the default palette for an entire run, use
  the command line "map=" parameter.

  The default filetype for color-map files is ".MAP".

  These color-maps are ASCII text files set up as a series of RGB triplet
  values (one triplet per line, encoded as the red, green, and blue [RGB]
  components of the color).

  Note that .MAP file color values are in GIF format - values go from 0
  (low) to 255 (high), so for a VGA adapter they get divided by 4 before
  being stuffed into the VGA's Video-DAC registers (so '6' and '7' end up
  referring to the same color value).

  Fractint is distributed with some sample .MAP files:
     ALTERN.MAP    the famous "Peterson-Vigneau Pseudo-Grey Scale"
     BLUES.MAP     for rainy days, by Daniel Egnor
     CHROMA.MAP    general purpose, chromatic
     DEFAULT.MAP   the VGA start-up values
     FIRESTRM.MAP  general purpose, muted fire colors
     GAMMA1.MAP and GAMMA2.MAP  Lee Crocker's response to ALTERN.MAP
     GLASSES1.MAP  used with 3d glasses modes
     GLASSES2.MAP  used with 3d glasses modes
     GOODEGA.MAP   for EGA users
     GREEN.MAP     shaded green
     GREY.MAP      another grey variant
     GRID.MAP      for stereo surface grid images
     HEADACHE.MAP  major stripes, by D. Egnor (try cycling and hitting <2>)
     LANDSCAP.MAP  Guruka Singh Khalsa's favorite map for plasma "landscapes"
     NEON.MAP      a flashy map, by Daniel Egnor
     PAINTJET.MAP  high resolution mode PaintJet colors
     ROYAL.MAP     the royal purple, by Daniel Egnor
     TOPO.MAP      Monte Davis's contribution to full color terrain
     VOLCANO.MAP   an explosion of lava, by Daniel Egnor


 3.3 Autokey Mode

  The autokey feature allows you to set up beautiful self-running demo
  "loops". You can set up hypnotic sequences to attract people to a booth,
  to generate sequences for special effects, to teach how Fractal
  exploring is done, etc.

  A sample autokey file (DEMO.KEY) and a batch to run it (DEMO.BAT) are
  included with Fractint. Type "demo" at the DOS prompt to run it.

  Autokey record mode is enabled with the command line parameter
  "AUTOKEY=RECORD". Keystrokes are saved in an intelligible text format in
  a file called AUTO.KEY. You can change the file name with the
  "AUTOKEYNAME=" parameter.

                     Fractint Version xx.xx                     Page 73

  Playback is enabled with the parameter "AUTOKEY=PLAY". Playback can be
  terminated by pressing the <Esc> key.

  After using record mode to capture an autokey file, you'll probably want
  to touch it up using your editor before playing it back.

  Separate lines are not necessary but you'll probably find it easier to
  understand an autokey file if you put each command on a separate line.
  Autokey files can contain the following:

    Quoted strings. Fractint reads whatever is between the quotes just as
    if you had typed it. For example,
        "t" "ifs" issues the "t" (type) command and then enters the
    letters i", "f", and "s" to select the ifs type.

    Symbols for function keys used to select a video mode. Examples:
        F3  -- Function key 3
        SF3 --<Shift> and <F3> together

    Special keys: ENTER ESC F1 PAGEUP PAGEDOWN HOME END LEFT RIGHT UP DOWN
    INSERT DELETE TAB

    WAIT <nnn.n> -- wait nnn.n seconds before continuing

    CALCWAIT -- pause until the current fractal calculation or file save
    or restore is finished. This command makes demo files more robust
    since calculation times depend on the  speed of the machine running
    the demo - a "WAIT 10" command may allow enough time to complete a
    fractal on one machine, but not on another. The record mode does not
    generate this command - it should be added by hand to the autokey file
    whenever there is a process that should be allowed to run to
    completion.

    GOTO target -- The autokey file continues to be read from the label
    "target". The label can be any word that does not duplicate a key
    word.  It must be present somewhere in the autokey file with a colon
    after it.  Example:
        MESSAGE 2 This is executed once
        start:
        MESSAGE 2 This is executed repeatedly
        GOTO start
    GOTO is mainly useful for writing continuous loop demonstrations. It
    can also be useful when debugging an autokey file, to skip sections of
    it.

    ; -- A semi-colon indicates that the rest of the line containing it is
    a comment.

    MESSAGE nn <Your message here> -- Places a message on the top of the
    screen for nn seconds

  Making Fractint demos can be tricky. Here are some suggestions which may
  help:

                     Fractint Version xx.xx                     Page 74

    Start Fractint with "fractint autokeyname=mydemo.key autokey=record".
    Use a unique name each time you run so that you don't overwrite prior
    files.

    When in record mode, avoid using the cursor keys to select filenames,
    fractal types, formula names, etc. Instead, try to type in names. This
    will ensure that the exact item you want gets chosen during playback
    even if the list is different then.

    Beware of video mode assumptions. It is safest to build a separate
    demo for different resolution monitors.

    When in the record mode, try to type names quickly, then pause. If you
    pause partway through a name Fractint will break up the string in the
    .KEY file. E.g. if you paused in the middle of typing fract001, you
    might get:
        "fract"
        WAIT 2.2
        "001"
    No harm done, but messy to clean up. Fractint ignores pauses less than
    about 1/2 second.

    DO pause when you want the viewer to see what is happening during
    playback.

    When done recording, clean up your mydemo.key file. Insert a CALCWAIT
    after each keystroke which triggers something that takes a variable
    amount of time (calculating a fractal, restoring a file, saving a
    file).

    Add comments with ";" to the file so you know what is going on in
    future.

    It is a good idea to use INSERT before a GOTO which restarts the demo.
    The <insert> key resets Fractint as if you exited the program and
    restarted it.

  Warning: an autokey file built for this version of Fractint will
  probably require some retouching before it works with future releases of
  Fractint.  We have no intention of making sure that the same sequence of
  keystrokes will have exactly the same effect from one version of
  Fractint to the next. That would require pretty much freezing Fractint
  development, and we just love to keep enhancing it!


 3.4 Distance Estimator Method

  This is Phil Wilson's implementation of an alternate method for the M
  and J sets, based on work by mathematician John Milnor and described in
  "The Science of Fractal Images", p. 198.  While it can take full
  advantage of your color palette, one of the best uses is in preparing
  monochrome images for a printer.  Using the 1600x1200x2 disk-video mode
  and an HP LaserJet, we have produced pictures of quality equivalent to
  the black and white illustrations of the M-set in "The Beauty of
  Fractals."

                     Fractint Version xx.xx                     Page 75

  The distance estimator method widens very thin "strands" which are part
  of the "inside" of the set.  Instead of hiding invisibly between pixels,
  these strands are made one pixel wide.

  Though this option is available with any escape time fractal type, the
  formula used is specific to the mandel and julia types - for most other
  types it doesn't do a great job.

  To turn on the distance estimator method with any escape time  fractal
  type, set the "Distance Estimator" value on the <Y> options screen (or
  use the "distest=" command line parameter).

  Setting the distance estimator option to a negative value -nnn enables
  edge-tracing mode.  The edge of the set is display as color number nnn.
  This option works best when the "inside" and "outside" color values are
  also set to some other value(s).

  In a 2 color (monochrome) mode, setting to any positive value results in
  the inside of the set being expanded to include edge points, and the
  outside points being displayed in the other color.

  In color modes, setting to value 1 causes the edge points to be
  displayed using the inside color and the outside points to be displayed
  in their usual colors.  Setting to a value greater than one causes the
  outside points to be displayed as contours, colored according to their
  distance from the inside of the set.  Use a higher value for narrower
  color bands, a lower value for wider ones.  1000 is a good value to
  start with.

  The second distance estimator parameter ("width factor") sets the
  distance from the inside of the set which is to be considered as part of
  the inside.  This value is expressed as a percentage of a pixel width,
  the default is 71.  Negative values are now allowed and give a fraction
  of a percent of the pixel width.  For example: -71 gives 1/71 % of the
  pixel width.

  You should use 1 or 2 pass mode with the distance estimator method, to
  avoid missing some of the thin strands made visible by it.  For the
  highest quality, "maxiter" should also be set to a high value, say 1000
  or so.  You'll probably also want "inside" set to zero, to get a black
  interior.

  Enabling the distance estimator method automatically toggles to floating
  point mode. When you reset distest back to zero, remember to also turn
  off floating point mode if you want it off.

  Unfortunately, images using the distance estimator method can take many
  hours to calculate even on a fast machine with a coprocessor, especially
  if a high "maxiter" value is used.  One way of dealing with this is to
  leave it turned off while you find and frame an image.  Then hit <B> to
  save the current image information in a parameter file (see Parameter
  Save/Restore Commands (p. 23)).  Use an editor to change the parameter
  file entry, adding "distest=1", "video=something" to select a high-
  resolution monochrome disk-video mode, "maxiter=1000", and "inside=0".
  Run the parameter file entry with the <@> command when you won't be
  needing your machine for a while (over the weekend?)

                     Fractint Version xx.xx                     Page 76

 3.5 Inversion

  Many years ago there was a brief craze for "anamorphic art": images
  painted and viewed with the use of a cylindrical mirror, so that  they
  looked weirdly distorted on the canvas but correct in the distorted
  reflection. (This byway of art history may be a useful defense when your
  friends and family give you odd looks for staring at fractal images
  color-cycling on a CRT.)

  The Inversion option performs a related transformation on most of the
  fractal types. You define the center point and radius of a circle;
  Fractint maps each point inside the circle to a corresponding point
  outside, and vice-versa. This is known to mathematicians as inverting
  (or if you want to get precise, "everting") the plane, and is something
  they can contemplate without getting a headache. John Milnor (also
  mentioned in connection with the Distance Estimator Method (p. 74)),
  made his name in the 1950s with a method for everting a seven-
  dimensional sphere, so we have a lot of catching up to do.

  For example, if a point inside the circle is 1/3 of the way from the
  center to the radius, it is mapped to a point along the same radial
  line, but at a distance of (3 * radius) from the origin. An outside
  point at 4 times the radius is mapped inside at 1/4 the radius.

  The inversion parameters on the <Y> options screen allow entry of the
  radius and center coordinates of the inversion circle. A default choice
  of -1 sets the radius at 1/6 the smaller dimension of the image
  currently on the screen.  The default values for Xcenter and Ycenter use
  the coordinates currently mapped to the center of the screen.

  Try this one out with a Newton (p. 38) plot, so its radial "spokes"
  will give you something to hang on to. Plot a Newton-method image, then
  set the inversion radius to 1, with default center coordinates. The
  center "explodes" to the periphery.

  Inverting through a circle not centered on the origin produces bizarre
  effects that we're not even going to try to describe. Aren't computers
  wonderful?


 3.6 Decomposition

  You'll remember that most fractal types are calculated by iterating a
  simple function of a complex number, producing another complex number,
  until either the number exceeds some pre-defined "bailout" value, or the
  iteration limit is reached. The pixel corresponding to the starting
  point is then colored based on the result of that calculation.

  The decomposition option ("decomp=", on the <X> screen) toggles to
  another coloring protocol.  Here the points are colored according to
  which quadrant of the complex plane (negative real/positive imaginary,
  positive real/positive imaginary, etc.) the final value is in. If you
  use 4 as the parameter, points ending up in each quadrant are given
  their own color; if 2 (binary decomposition), points in alternating
  quadrants are given 2 alternating colors.

                     Fractint Version xx.xx                     Page 77

  The result is a kind of warped checkerboard coloring, even in areas that
  would ordinarily be part of a single contour. Remember, for the M-set
  all points whose final values exceed 2 (by any amount) after, say, 80
  iterations are normally the same color; under decomposition, Fractint
  runs [bailout-value] iterations and then colors according to where the
  actual final value falls on the complex plane.

  When using decomposition, a higher bailout value will give a more
  accurate plot, at some expense in speed.  You might want to set the
  bailout value (in the parameters prompt following selection of a new
  fractal type; present for most but not all types) to a higher value than
  the default.  A value of about 50 is a good compromise for M/J sets.


 3.7 Logarithmic Palettes and Color Ranges

  By default, Fractint maps iterations to colors 1:1. I.e. if the
  calculation for a fractal "escapes" (exceeds the bailout value) after N
  iterations, the pixel is colored as color number N. If N is greater than
  the number of colors available, it wraps around. So, if you are using a
  16-color video mode, and you are using the default maximum iteration
  count of 150, your image will run through the 16-color palette 150/16 =
  9.375 times.

  When you use Logarithmic palettes, the entire range of iteration values
  is compressed to map to one span of the color range.  This results in
  spectacularly different images if you are using a high iteration limit
  near the current iteration maximum of 32000 and are zooming in on an
  area near a "lakelet".

  When using a compressed palette in a 256 color mode, we suggest changing
  your colors from the usual defaults.  The last few colors in the default
  IBM VGA color map are black.  This results in points nearest the "lake"
  smearing into a single dark band, with little contrast from the blue (by
  default) lake.

  Fractint has a number of types of compressed palette, selected by the
  "Log Palette" line on the <X> screen, or by the "logmap=" command line
  parameter:

    logmap=1: for standard logarithmic palette.

    logmap=-1: "old" logarithmic palette. This variant was the only one
    used before Fractint 14.0. It differs from logmap=1 in that some
    colors are not used - logmap=1 "spreads" low color numbers which are
    unused by logmap=-1's pure logarithmic mapping so that all colors are
    assigned.

    logmap=N (>1): Same as logmap=1, but starting from iteration count N.
    Pixels with iteration counts less than N are mapped to color 1. This
    is useful when zooming in an area near the lake where no points in the
    image have low iteration counts - it makes use of the low colors which
    would otherwise be unused.

                     Fractint Version xx.xx                     Page 78

    logmap=-N (<-1): Similar to logmap=N, but uses a square root
    distribution of the colors instead of a logarithmic one.

    logmap=2 or -2: Auto calculates the logmap value for maximum effect.

  Another way to change the 1:1 mapping of iteration counts to colors is
  to use the "RANGES=" parameter.  It has the format:
     RANGES=aa/bb/cc/dd/...

  Iteration counts up to and including the first value are mapped to color
  number 0, up to and including the second value to color number 1, and so
  on. The values must be in ascending order.

  A negative value can be specified for "striping". The negative value
  specifies a stripe width, the value following it specifies the limit of
  the striped range.  Two alternating colors are used within the striped
  range.

  Example:
     RANGES=0/10/30/-5/65/79/32000
  This example maps iteration counts to colors as follows:

      color    iterations
      -------------------
        0      unused (formula always iterates at least once)
        1       1 to 10
        2      11 to 30
        3      31 to 35, 41 to 45, 51 to 55, and 61 to 65
        4      36 to 40, 46 to 50, and 56 to 60
        5      66 to 79
        6      80 and greater
  Note that the maximum value in a RANGES parameter is 32767.


 3.8 Biomorphs

  Related to Decomposition (p. 76) are the "biomorphs" invented by
  Clifford Pickover, and discussed by A. K. Dewdney in the July 1989
  "Scientific American", page 110.  These are so-named because this
  coloring scheme makes many fractals look like one-celled animals.  The
  idea is simple.  The escape-time algorithm terminates an iterating
  formula when the size of the orbit value exceeds a predetermined bailout
  value. Normally the pixel corresponding to that orbit is colored
  according to the iteration when bailout happened. To create biomorphs,
  this is modified so that if EITHER the real OR the imaginary component
  is LESS than the bailout, then the pixel is set to the "biomorph" color.
  The effect is a bit better with higher bailout values: the bailout is
  automatically set to 100 when this option is in effect. You can try
  other values with the "bailout=" option.

  The biomorph option is turned on via the "biomorph=nnn" command-line
  option (where "nnn" is the color to use on the affected pixels).  When
  toggling to Julia sets, the default corners are three times bigger than
  normal to allow seeing the biomorph appendages. Does not work with all
  types - in particular it fails with any of the mandelsine family.
  However, if you are stuck with monochrome graphics, try it - works great

                     Fractint Version xx.xx                     Page 79

  in two-color modes. Try it with the marksmandel and marksjulia types.


 3.9 Continuous Potential

  Note: This option can only be used with 256 color modes.

  Fractint's images are usually calculated by the "level set" method,
  producing bands of color corresponding to regions where the calculation
  gives the same value. When "3D" transformed (see "3D" Images (p. 85)),
  most images other than plasma clouds are like terraced landscapes: most
  of the surface is either horizontal or vertical.

  To get the best results with the "illuminated" 3D fill options 5 and 6,
  there is an alternative approach that yields continuous changes in
  colors.

  Continuous potential is approximated by calculating

           potential =  log(modulus)/2^iterations

  where "modulus" is the orbit value (magnitude of the complex number)
  when the modulus bailout was exceeded, at the "iterations" iteration.
  Clear as mud, right?

  Fortunately, you don't have to understand all the details. However,
  there ARE a few points to understand. First, Fractint's criterion for
  halting a fractal calculation, the "modulus bailout value", is generally
  set to 4.  Continuous potential is inaccurate at such a low value.

  The bad news is that the integer math which makes the "mandel" and
  "julia" types so fast imposes a hard-wired maximum value of 127. You can
  still make interesting images from those types, though, so don't avoid
  them. You will see "ridges" in the "hillsides." Some folks like the
  effect.

  The good news is that the other fractal types, particularly the
  (generally slower) floating point algorithms, have no such limitation.
  The even better news is that there is a floating-point algorithm for the
  "mandel" and "julia" types.  To force the use of a floating-point
  algorithm, use Fractint with the "FLOAT=YES" command-line toggle.  Only
  a few fractal types like plasma clouds, the Barnsley IFS type, and
  "test" are unaffected by this toggle.

  The parameters for continuous potential are:
      potential=maxcolor[/slope[/modulus[/16bit]]]
  These parameters are present on the <Y> options screen.

  "Maxcolor" is the color corresponding to zero potential, which plots as
  the TOP of the mountain. Generally this should be set to one less than
  the number of colors, i.e. usually 255. Remember that the last few
  colors of the default IBM VGA palette are BLACK, so you won't see what
  you are really getting unless you change to a different palette.

                     Fractint Version xx.xx                     Page 80

  "Slope" affects how rapidly the colors change -- the slope of the
  "mountains" created in 3D. If this is too low, the palette will not
  cover all the potential values and large areas will be black. If it is
  too high, the range of colors in the picture will be much less than
  those available.  There is no easy way to predict in advance what this
  value should be.

  "Modulus" is the bailout value used to determine when an orbit has
  "escaped". Larger values give more accurate and smoother potential. A
  value of 500 gives excellent results. As noted, this value must be <128
  for the integer fractal types (if you select a higher number, they will
  use 127).

  "16bit":  If you transform a continuous potential image to 3D, the
  illumination modes 5 and 6 will work fine, but the colors will look a
  bit granular. This is because even with 256 colors, the continuous
  potential is being truncated to integers. The "16bit" option can be used
  to add an extra 8 bits of goodness to each stored pixel, for a much
  smoother result when transforming to 3D.

  Fractint's visible behavior is unchanged when 16bit is enabled, except
  that solid guessing and boundary tracing are not used. But when you save
  an image generated with 16bit continuous potential:
   o The saved file is a fair bit larger.
   o Fractint names the file with a .POT extension instead of .GIF, if you
     didn't specify an extension in "savename".
   o The image can be used as input to a subsequent <3> command to get the
     promised smoother effect.
   o If you happen to view the saved image with a GIF viewer other than
     Fractint, you'll find that it is twice as wide as it is supposed to
     be. (Guess where the extra goodness was stored!) Though these files
     are structurally legal GIF files the double-width business made us
     think they should perhaps not be called GIF - hence the .POT filename
     extension.

  A 16bit (.POT) file can be converted to an ordinary 8 bit GIF by
  <R>estoring it, changing "16bit" to "no" on the <Y> options screen, and
  <S>aving.

  You might find with 16bit continuous potential that there's a long delay
  at the start of an image, and disk activity during calculation. Fractint
  uses its disk-video cache area to store the extra 8 bits per pixel - if
  there isn't sufficient memory available, the cache will page to disk.

  The following commands can be used to recreate the image that Mark
  Peterson first prototyped for us, and named "MtMand":

          TYPE=mandel
          CORNERS=-0.19920/-0.11/1.0/1.06707
          INSIDE=255
          MAXITER=255
          POTENTIAL=255/2000/1000/16bit
          PASSES=1
          FLOAT=yes

                     Fractint Version xx.xx                     Page 81

  Note that prior to version 15.0, Fractint:
   o Produced "16 bit TGA potfiles" This format is no longer generated,
     but you can still (for a release or two) use <R> and <3> with those
     files.
   o Assumed "inside=maxit" for continuous potential. It now uses the
     current "inside=" value - to recreate prior results you must be
     explicit about this parameter.


 3.10 Starfields

  Once you have generated your favorite fractal image, you can convert it
  into a fractal starfield with the 'a' transformation (for 'astronomy'? -
  once again, all of the good letters were gone already).  Stars are
  generated on a pixel-by-pixel basis - the odds that a particular pixel
  will coalesce into a star are based (partially) on the color index of
  that pixel.

  (The following was supplied by Mark Peterson, the starfield author).

  If the screen were entirely black and the 'Star Density per Pixel' were
  set to 30 then a starfield transformation would create an evenly
  distributed starfield with an average of one star for every 30 pixels.

  If you're on a 320x200 screen then you have 64000 pixels and would end
  up with about 2100 stars.  By introducing the variable of 'Clumpiness'
  we can create more stars in areas that have higher color values.  At
  100% Clumpiness a color value of 255 will change the average of finding
  a star at that location to 50:50.  A lower clumpiness values will lower
  the amount of probability weighting.  To create a spiral galaxy draw
  your favorite spiral fractal (IFS, Julia, or Mandelbrot) and perform a
  starfield transformation.  For general starfields I'd recommend
  transforming a plasma fractal.

  Real starfields have many more dim stars than bright ones because very
  few stars are close enough to appear bright.  To achieve this effect the
  program will create a bell curve based on the value of ratio of Dim
  stars to bright stars.  After calculating the bell curve the curve is
  folded in half and the peak used to represent the number of dim stars.

  Starfields can only be shown in 256 colors.  Fractint will automatically
  try to load ALTERN.MAP and abort if the map file cannot be found.


 3.11 Bailout Test

  The bailout test is used to determine if we should stop iterating before
  the maximum iteration count is reached.  This test compares the value
  determined by the test to the "bailout" value set via the <Z> screen.
  The default bailout test compares the magnitude or modulus of a complex
  variable to some bailout value:

  bailout test = |z| = sqrt(x^2 + y^2) >= 2

                     Fractint Version xx.xx                     Page 82

  As a computational speedup, we square both sides of this equation and the
  bailout test used by Fractint is:

  bailout test = |z|^2 = x^2 + y^2 >= 4

  Using a "bailout" other than 4 allows us to change when the bailout will occur.

  The following bailout tests have been implemented on the <Z> screen:

     mod:     x^2 + y^2 >= bailout

     real:    x^2       >= bailout

     imag:    y^2       >= bailout

     or:      x^2 >= bailout  or   y^2 >= bailout

     and:     x^2 >= bailout  and  y^2 >= bailout

  The bailout test feature has not been implemented for all applicable
  fractal types.  This is due to the speedups used for these types.  Some
  of these bailout tests show the limitations of the integer math routines
  by clipping the spiked ends off of the protrusions.


 3.12 Random Dot Stereograms (RDS)

  Random Dot Stereograms (RDS) are a way of encoding stereo images on a
  flat screen. Fractint can convert any image to a RDS using either the
  color number in the current palette or the grayscale value as depth. Try
  these steps. Generate a plasma fractal using the 640x480x256 video mode.
  When the image on the screen is complete, press <ctrl-s> ("s" for
  "Stereo"), and press <Enter> at the "RDS Parameters" screen prompt to
  accept the defaults. (More on the parameters in a moment.) The screen
  will be converted into a seemingly random collection of colored dots.
  Relax your eyes, looking through the screen rather than at the screen
  surface. The image will (hopefully) resolve itself into the hills and
  valleys of the 3D Plasma fractal.

  Because pressing the two-keyed <ctrl-s> gets tiresome after a while, we
  have made <k> key a synonym for <ctrl-s> for convenience.  Don't get too
  attached to <k> though; we reserve the right to reuse it for another
  purpose later.

  The RDS feature has five and sometimes six parameters. Pressing <ctrl-s>
  always takes you to the parameter screen.

  The first parameter allows you to control the depth effect. A larger
  value (positive or negative) exaggerates the sense of depth. If you make
  the depth negative, the high and low areas of the image are reversed. If
  your RDS image is streaky try either a lower depth factor or a higher
  resolution.

  The second parameter indicates the overall width in inches of the image
  on your monitor or printout. The default value of 10 inches is roughly
  the width of an image on a standard 14" to 16" monitor. This value does

                     Fractint Version xx.xx                     Page 83

  not normally need to be changed for viewing images on standard monitors.
  However, if your monitor or image hardcopy is much wider or narrower
  than 10 inches (25 cm), and you have trouble seeing the image, enter the
  image width in inches. The issue here is that if the widest separation
  of left and right pixels is greater than the physical separation of your
  eyes, you will not be able to fuse the images. Conversely, a too-small
  separation may cause your eyes to hyper-converge (fuse the wrong pixels
  together). A larger width value reduces the width between left and right
  pixels. You can use the calibration feature to help set the width
  parameter - see below. Once you have found a good width setting, you can
  place the value in your SSTOOLS.INI file with the command
  monitorwidth=<nnn>.

  The third parameter allows you to control the method use to extract
  depth information from the original image. If your answer "no" at the
  "Use Grayscale value for Depth" prompt, then the color number of each
  pixel will be used.  This value is independent of active color palette.
  If you answer "yes" and the prompt, then the depth values are keyed to
  the brightness of the color, which will change if you change palettes.

  The fourth parameter allows you to set the position of vertical stereo
  calibration bars to the middle or the top of the image, or have the bars
  initially turned off. Use this feature to help you adjust your eye's
  convergence to see the image. You will see two vertical bars on the
  screen.  You can turn off and on these bars with the <Enter> or <Space>
  keys after generating the RDS image. If you save an RDS image by
  pressing <s>, if the bars are turned on at the time, they become a
  permanent part of the image.

  As you relax your eyes and look past the screen, these bars will appear
  as four bars. When you adjust your eyes so that the two middle bars
  merge into one bar, the 3D image should appear. The bars are set for the
  average depth in the area near the bars. They should always be closer
  together than the physical separation of your eyes, but not much less
  than about 1.5 inches.  About 1.75 inches is ideal for many images. The
  depth and screen width controls affect the width of the bars.

  At the RDS Parameters screen, you can select bars at the middle of the
  screen or the top. If you select "none", the bars will initially be off,
  but immediately after generation of the image you can still turn on the
  bars with <Enter> or <Space> before you press any other keys. If the
  initial setting of the calibration bars is "none", then if the bars are
  turned on later they will appear in the middle. Hint: if you cycle the
  colors and find you can't see the calibration bar, press <Enter> or
  <Space> twice, and the bars will turn to a more visible color.

  The fifth parameter asks if you want to use an image map GIF file
  instead of using random dots. An image map can give your RDS image a
  more interesting background texture than the random dots. If you answer
  "yes" at the Use image map? prompt, Fractint will present you with a
  file selection list of GIF images. Fractint will then go ahead and
  transform your original image to RDS using the selected image map to
  provide the "random" dots.

                     Fractint Version xx.xx                     Page 84

  After you have selected an image map file, the next time you reach the
  RDS Parameters screen you will see an additional prompt asking if you
  want to use the same image map file again. Answering "yes" avoids the
  file selection menu.

  The best images to use as image maps are detailed textures with no solid
  spots. The default type=circle fractal works well, as do the barnsley
  fractals if you zoom in a little way. If the image map is smaller than
  your RDS image, the image map will repeated to fill the space. If the
  image map is larger, just the upper left corner of the image map will be
  used.

  The original image you are using for your stereogram is saved, so if you
  want to modify the stereogram parameters and try again, just press
  <ctrl-s> (or <k>) to get the parameter screen, changes the parameters,
  and press <Enter>. The original image is restored and an RDS transform
  with the revised parameters is performed. If you press <s> when viewing
  an RDS image, after the RDS image is saved, the original is restored.

  Try the RDS feature with continuous potential Mandelbrots as well as
  plasma fractals.

  For a summary of keystrokes in RDS mode, see RDS Commands (p. 30)

                     Fractint Version xx.xx                     Page 85

 4. "3D" Images

  Fractint can restore images in "3D". Important: we use quotation marks
  because it does not CREATE images of 3D fractal objects (there are such,
  but we're not there yet.) Instead, it restores .GIF images as a 3D
  PROJECTION or STEREO IMAGE PAIR.  The iteration values you've come to
  know and love, the ones that determine pixel colors, are translated into
  "height" so that your saved screen becomes a landscape viewed in
  perspective. You can even wrap the landscape onto a sphere for
  realistic-looking planets and moons that never existed outside your PC!

  We suggest starting with a saved plasma-cloud screen. Hit <3> in main
  command mode to begin the process. Next, select the file to be
  transformed, and the video mode. (Usually you want the same video mode
  the file was generated in; other choices may or may not work.)

  After hitting <3>, you'll be bombarded with a long series of options.
  Not to worry: all of them have defaults chosen to yield an acceptable
  starting image, so the first time out just pump your way through with
  the <Enter> key. When you enter a different value for any option, that
  becomes the default value the next time you hit <3>, so you can change
  one option at a time until you get what you want. Generally <ESC> will
  take you back to the previous screen.

  Once you're familiar with the effects of the 3D option values you have a
  variety of options on how to specify them. You can specify them all on
  the command line (there ARE a lot of them so they may not all fit within
  the DOS command line limits), with an SSTOOLS.INI file, or with a
  parameter file.

  Here's an example for you power FRACTINTers, the command

        FRACTINT MYFILE SAVENAME=MY3D 3D=YES BATCH=YES

  would make Fractint load MYFILE.GIF, re-plot it as a 3D landscape
  (taking all of the defaults), save the result as MY3D.GIF, and exit to
  DOS. By the time you've come back with that cup of coffee, you'll have a
  new world to view, if not conquer.

  Note that the image created by 3D transformation is treated as if it
  were a plasma cloud - We have NO idea how to retain the ability to zoom
  and pan around a 3D image that has been twisted, stretched, perspective-
  ized, and water-leveled. Actually, we do, but it involves the kind of
  hardware that Industrial Light & Magic, Pixar et al. use for feature
  films. So if you'd like to send us a check equivalent to George Lucas'
  net from the "Star Wars" series...


 4.1 3D Mode Selection

  After hitting <3> and getting past the filename prompt and video mode
  selection, you're presented with a "3d Mode Selection" screen.  If you
  wish to change the default for any of the following parameters, use the
  cursor keys to move through the menu. When you're satisfied press
  <Enter>.

                     Fractint Version xx.xx                     Page 86

  Preview Mode: Preview mode provides a rapid look at your transformed
     image using by skipping a lot of rows and filling the image in. Good
     for quickly discovering the best parameters. Let's face it, the
     Fractint authors most famous for "blazingly fast" code *DIDN'T* write
     the 3D routines!  [Pieter: "But they *are* picking away it and making
     some progress in each release."]

  Show Box: If you have selected Preview Mode you have another option to
     worry about. This is the option to show the image box in scaled and
     rotated coordinates x, y, and z. The box only appears in rectangular
     transformations and shows how the final image will be oriented. If
     you select light source in the next screen, it will also show you the
     light source vector so you can tell where the light is coming from in
     relation to your image. Sorry no head or tail on the vector yet.

  Coarseness: This sets how many divisions the image will be divided into
     in the y direction, if you select preview mode, ray tracing output,
     or grid fill in the "Select Fill Type" screen.

  Spherical Projection: The next question asks if you want a sphere
     projection. This will take your image and map it onto a plane if you
     answer "no" or a sphere if you answer "yes" as described above. Try
     it and you'll see what we mean.  See Spherical Projection (p. 93).

  Stereo:

     Stereo sound in Fractint? Well, not yet. Fractint now allows you to
     create 3D images for use with red/blue glasses like 3D comics you may
     have seen, or images like Captain EO.

     Option 0 is normal old 3D you can look at with just your eyes.

     Options 1 and 2 require the special red/blue-green glasses.  They are
     meant to be viewed right on the screen or on a color print off of the
     screen. The image can be made to hover entirely or partially in front
     of the screen. Great fun!  These two options give a gray scale image
     when viewed.

     Option 1 gives 64 shades of gray but with half the spatial resolution
     you have selected. It works by writing the red and blue images on
     adjacent pixels, which is why it eats half your resolution. In
     general, we recommend you use this only with resolutions above
     640x350. Use this mode for continuous potential landscapes where you
     *NEED* all those shades.

     Option "2" gives you full spatial resolution but with only 16 shades
     of gray. If the red and blue images overlap, the colors are mixed.
     Good for wire-frame images (we call them surface grids), lorenz3d and
     3D IFS. Works fine in 16 color modes.

     Option 3 is for creating stereo pair images for view later with more
     specialized equipment. It allows full color images to be presented in
     glorious stereo. The left image presented on the screen first. You
     may photograph it or save it. Then the second image is presented, you
     may do the same as the first image. You can then take the two images
     and convert them to a stereo image pair as outlined by Bruce Goren

                     Fractint Version xx.xx                     Page 87

     (see below).

     Also see Stereo 3D Viewing (p. 89).

  Ray Tracing Output:

     Fractint can create files of its 3d transformations which are
     compatible with many ray tracing programs. Currently four are
     supported directly: DKB (now obsolete), VIVID, MTV, and RAYSHADE. In
     addition a "RAW" output is supported which can be relatively easily
     transformed to be usable by many other products.  One other option is
     supported: ACROSPIN.  This is not a ray tracer, but the same Fractint
     options apply - see Acrospin (p. 171).

     Option values:
        0  disables the creation of ray tracing output
        1  DKB format (obsolete-see below)
        2  VIVID format
        3  generic format (must be massaged externally)
        4  MTV format
        5  RAYSHADE format
        6  ACROSPIN format
     Users of POV-Ray can use the DKB output and convert to POV-Ray with
     the DKB2POV utility that comes with POV-Ray. A better (faster)
     approach is to create a RAW output file and convert to POV-Ray with
     RAW2POV.  A still better approach is to use POV-Ray's height field
     feature to directly read the fractal .GIF or .POT file and do the 3D
     transformation inside POV-Ray.

     All ray tracing files consist of triangles which follow the surface
     created by Fractint during the 3d transform. Triangles which lie
     below the "water line" are not created in order to avoid causing
     unnecessary work for the poor ray tracers which are already
     overworked.  A simple plane can be substituted by the user at the
     waterline if needed.

     The size (and therefore the number) of triangles created is
     determined by the "coarse" parameter setting. While generating the
     ray tracing file, you will view the image from above and watch it
     partitioned into triangles.

     The color of each triangle is the average of the color of its
     verticies in the original image, unless BRIEF is selected.

     If BRIEF is selected, a default color is assigned at the begining of
     the file and is used for all triangles.

     Also see Interfacing with Ray Tracing Programs (p. 96).

  Brief output:

     This is a ray tracing sub-option.  When it is set to yes, Fractint
     creates a considerably smaller and somewhat faster file. In this
     mode, all triangles use the default color specified at the begining
     of the file.  This color should be edited to supply the color of your
     choice.

                     Fractint Version xx.xx                     Page 88

  Targa Output:

     If you want any of the 3d transforms you select to be saved as a
     Targa-24 file or overlayed onto one, select yes for this option.  The
     overlay option in the final screen determines whether you will create
     a new file or overlay an existing one.

  MAP File name:

     Imediately after selecting the previous options, you will be given
     the chance to select an alternate color MAP file. The default is to
     use the current MAP. If you want another MAP used, then enter your
     selection at this point.

  Output File Name:

     This is a ray tracing sub-option, used to specify the name of the
     file to be written.  The default name is FRACT001.RAY.  The name is
     incremented by one each time a file is written.  If you have not set
     "overwrite=yes" then the file name will also be automatically
     incremented to avoid over-writing previous files.

  When you are satisfied with your selections press enter to go to the
  next parameter screen.


 4.2 Select Fill Type Screen

  This option exists because in the course of the 3D projection, portions
  of the original image may be stretched to fit the new surface. Points of
  an image that formerly were right next to each other, now may have a
  space between them. This option generally determines what to do with the
  space between the mapped dots. It is not used if you have selected a
  value for RAY other than 0.

  For an illustration, pick the second option "just draw the points",
  which just maps points to corresponding points. Generally this will
  leave empty space between many of the points. Therefore you can choose
  various algorithms that "fill in" the space between the points in
  various ways.

  Later, try the first option "make a surface grid." This option will make
  a grid of the surface which is as many divisions in the original "y"
  direction as was set in "coarse" in the first screen. It is very fast,
  and can give you a good idea what the final relationship of parts of
  your picture will look like.

  Later, try the second option "connect the dots (wire frame)", then
  "surface fills" - "colors interpolated" and "colors not interpolated",
  the general favorites of the authors. Solid fill, while it reveals the
  pseudo-geology under your pseudo-landscape, inevitably takes longer.

  Later, try the light source fill types. These two algorithms allow you
  to position the "sun" over your "landscape." Each pixel is colored
  according to the angle the surface makes with an imaginary light source.
  You will be asked to enter the three coordinates of the vector pointing

                     Fractint Version xx.xx                     Page 89

  toward the light in a following parameter screen - see Light Source
  Parameters (p. 92).

  "Light source before transformation" uses the illumination direction
  without transforming it. The light source is fixed relative to your
  computer screen.  If you generate a sequence of images with progressive
  rotation, the effect is as if you and the light source are fixed and the
  object is rotating. Therefore as the object rotates features of the
  object move in and out of the light.  This fill option was incorrect
  prior to version 16.1, and has been changed.

  "Light source after transformation" applies the same transformation to
  both the light direction and the object. Since both the light direction
  and the object are transformed, if you generate a sequence of images
  with the rotation progressively changed, the effect is as if the image
  and the light source are fixed in relation to each other and you orbit
  around the image. The illumination of features on the object is
  constant, but you see the object from different angles. This fill option
  was correct in earlier Fractint versions and has not been changed.

  For ease of discussion we will refer to the following fill types by
  these numbers:
      1 - surface grid
      2 - (default) - no fill at all - just draw the dots
      3 - wire frame - joins points with lines
      4 - surface fill - (colors interpolated)
      5 - surface fill - (interpolation turned off)
      6 - solid fill - draws lines from the "ground" up to the point
      7 - surface fill with light model - calculated before 3D transforms
      8 - surface fill with light model - calculated after 3D transforms

  Types 4, 7, and 8 interpolate colors when filling, making a very smooth
  fill if the palette is continuous. This may not be desirable if the
  palette is not continuous. Type 5 is the same as type 4 with
  interpolation turned off. You might want to use fill type 5, for
  example, to project a .GIF photograph onto a sphere. With type 4, you
  might see the filled-in points, since chances are the palette is not
  continuous; type 5 fills those same points in with the colors of
  adjacent pixels. However, for most fractal images, fill type 4 works
  better.

  This screen is not available if you have selected a ray tracing option.


 4.3 Stereo 3D Viewing

  The "Funny Glasses" (stereo 3D) parameter screen is presented only if
  you select a non-zero stereo option in the prior 3D parameters.  (See 3D
  Mode Selection (p. 85).)  We suggest you definitely use defaults at
  first on this screen.

  When you look at an image with both eyes, each eye sees the image in
  slightly different perspective because they see it from different
  places.

                     Fractint Version xx.xx                     Page 90

  The first selection you must make is ocular separation, the distance the
  between the viewers eyes. This is measured as a % of screen and is an
  important factor in setting the position of the final stereo image in
  front of or behind the CRT Screen.

  The second selection is convergence, also as a % of screen. This tends
  to move the image forward and back to set where it floats. More positive
  values move the image towards the viewer. The value of this parameter
  needs to be set in conjunction with the setting of ocular separation and
  the perspective distance. It directly adjusts the overall separation of
  the two stereo images. Beginning anaglyphers love to create images
  floating mystically in front of the screen, but grizzled old 3D veterans
  look upon such antics with disdain, and believe the image should be
  safely inside the monitor where it belongs!

  Left and Right Red and Blue image crop (% of screen also) help keep the
  visible part of the right image the same as the visible part of the left
  by cropping them. If there is too much in the field of either eye that
  the other doesn't see, the stereo effect can be ruined.

  Red and Blue brightness factor. The generally available red/blue-green
  glasses, made for viewing on ink on paper and not the light from a CRT,
  let in more red light in the blue-green lens than we would like. This
  leaves a ghost of the red image on the blue-green image (definitely not
  desired in stereo images). We have countered this by adjusting the
  intensity of the red and blue values on the CRT. In general you should
  not have to adjust this.

  The final entry is Map file name (present only if stereo=1 or stereo=2
  was selected).  If you have a special map file you want to use for
  Stereo 3D this is the place to enter its name. Generally glasses1.map is
  for type 1 (alternating pixels), and glasses2.map is for type 2
  (superimposed pixels). Grid.map is great for wire-frame images using 16
  color modes.

  This screen is not available if you have selected a ray tracing option.


 4.4 Rectangular Coordinate Transformation

  The first entries are rotation values around the X, Y, and Z axes. Think
  of your starting image as a flat map: the X value tilts the bottom of
  your monitor towards you by X degrees, the Y value pulls the left side
  of the monitor towards you, and the Z value spins it counter-clockwise.
  Note that these are NOT independent rotations: the image is rotated
  first along the X-axis, then along the Y-axis, and finally along the Z-
  axis. Those are YOUR axes, not those of your (by now hopelessly skewed)
  monitor. All rotations actually occur through the center of the original
  image. Rotation parameters are not used when a ray tracing option has
  been selected.

  Then there are three scaling factors in percent. Initially, leave the X
  and Y axes alone and play with Z, now the vertical axis, which
  translates into surface "roughness."  High values of Z make spiky, on-
  beyond-Alpine mountains and improbably deep valleys; low values make
  gentle, rolling terrain. Negative roughness is legal: if you're doing an

                     Fractint Version xx.xx                     Page 91

  M-set image and want Mandelbrot Lake to be below the ground, instead of
  eerily floating above, try a roughness of about -30%.

  Next we need a water level -- really a minimum-color value that performs
  the function "if (color < waterlevel) color = waterlevel". So it plots
  all colors "below" the one you choose at the level of that color, with
  the effect of filling in "valleys" and converting them to "lakes."

  Now we enter a perspective distance, which you can think of as the
  "distance" from your eye to the image. A zero value (the default) means
  no perspective calculations, which allows use of a faster algorithm.
  Perspective distance is not available if you have selected a ray tracing
  option.

  For non-zero values, picture a box with the original X-Y plane of your
  flat fractal on the bottom, and your 3D fractal inside. A perspective
  value of 100% places your eye right at the edge of the box and yields
  fairly severe distortion, like a close view through a wide-angle lens.
  200% puts your eye as far from the front of the box as the back is
  behind.  300% puts your eye twice as far from the front of the box as
  the back is, etc. Try about 150% for reasonable results. Much larger
  values put you far away for even less distortion, while values smaller
  than 100% put you "inside" the box. Try larger values first, and work
  your way in.

  Next, you are prompted for two types of X and Y shifts (now back in the
  plane of your screen) that let you move the final image around if you'd
  like to re-center it. The first set, x and y shift with perspective,
  move the image and the effect changes the perspective you see. The
  second set, "x and y adjust without perspective", move the image but do
  not change perspective.  They are used just for positioning the final
  image on the screen. Shifting of any type is not available if you have
  selected a ray tracing option.


 4.5 3D Color Parameters

  You are asked for a range of "transparent" colors, if any. This option
  is most useful when using the 3D Overlay Mode (p. 93).  Enter the color
  range (minimum and maximum value) for which you do not want to overwrite
  whatever may already be on the screen. The default is no transparency
  (overwrite everything).

  Now, for the final option. This one will smooth the transition between
  colors by randomizing them and reduce the banding that occurs with some
  maps. Select the value of randomize to between 0 (for no effect) and 7
  (to randomize your colors almost beyond use). 3 is a good starting
  point.

  That's all for this screen. Press enter for these parameters and the
  next and final screen will appear (honestly!).

                     Fractint Version xx.xx                     Page 92

 4.6 Light Source Parameters

  This one deals with all the aspects of light source and Targa files.

  You must choose the direction of the light from the light source. This
  will be scaled in the x, y, and z directions the same as the image. For
  example, 1,1,3 positions the light to come from the lower right front of
  the screen in relation to the untransformed image. It is important to
  remember that these coordinates are scaled the same as your image. Thus,
  "1,1,1" positions the light to come from a direction of equal distances
  to the right, below and in front of each pixel on the original image.
  However, if the x,y,z scale is set to 90,90,30 the result will be from
  equal distances to the right and below each pixel but from only 1/3 the
  distance in front of the screen i.e.. it will be low in the sky, say,
  afternoon or morning.

  Then you are asked for a smoothing factor. Unless you used Continuous
  Potential (p. 79) when generating the starting image, the illumination
  when using light source fills may appear "sparkly", like a sandy beach
  in bright sun. A smoothing factor of 2 or 3 will allow you to see the
  large-scale shapes better.

  Smoothing is primarily useful when doing light source fill types with
  plasma clouds. If your fractal is not a plasma cloud and has features
  with sharply defined boundaries (e.g. Mandelbrot Lake), smoothing may
  cause the colors to run. This is a feature, not a bug. (A copyrighted
  response of [your favorite commercial software company here], used by
  permission.)

  The ambient option sets the minimum light value a surface has if it has
  no direct lighting at all. All light values are scaled from this value
  to white. This effectively adjusts the depth of the shadows and sets the
  overall contrast of the image.

  If you selected the full color option, you have a few more choices.  The
  next is the haze factor. Set this to make distant objects more hazy.
  Close up objects will have little effect, distant objects will have
  most.  0 disables the function. 100 is the maximum effect, the farthest
  objects will be lost in the mist. Currently, this does not really use
  distance from the viewer, we cheat and use the y value of the original
  image. So the effect really only works if the y-rotation (set earlier)
  is between +/- 30.

  Next, you can choose the name under which to save your Targa file. If
  you have a RAM disk handy, you might want to create the file on it, for
  speed.  So include its  full path name in this option. If you have not
  set "overwrite=yes" then the file name will be incremented to avoid
  over-writing previous files. If you are going to overlay an existing
  Targa file, enter its name here.

  Next, you may select the background color for the Targa file. The
  default background on the Targa file is sky blue. Enter the Red, Green,
  and Blue component for the background color you wish.

                     Fractint Version xx.xx                     Page 93

  Finally, absolutely the last option (this time we mean it): you can now
  choose to overlay an existing Targa-24, type 2, non mapped, top-to-
  bottom file, such as created by Fractint or PVRay. The Targa file
  specified above will be overlayed with new info just as a GIF is
  overlayed on screen. Note: it is not necessary to use the "O" overlay
  command to overlay Targa files.  The Targa_Overlay option must be set to
  yes, however.


  You'll probably want to adjust the final colors for monochrome fill
  types using light source via color cycling (p. 16).  Try one of the
  more continuous palettes (<F8> through <F10>), or load the GRAY palette
  with the <A>lternate-map command.

  Now, lie down for a while in a quiet room with a damp washcloth on your
  forehead. Feeling better? Good -- because it's time to go back almost to
  the top of the 3D options and just say yes to:


 4.7 Spherical Projection

  Picture a globe lying on its side, "north" pole to the right. (It's our
  planet, and we'll position it the way we like.) You will be mapping the
  X and Y axes of the starting image to latitude and longitude on the
  globe, so that what was a horizontal row of pixels follows a line of
  longitude.  The defaults exactly cover the hemisphere facing you, from
  longitude 180 degrees (top) to 0 degrees (bottom) and latitude -90
  (left) to latitude 90 (right). By changing them you can map the image to
  a piece of the hemisphere or wrap it clear around the globe.

  The next entry is for a radius factor that controls the over-all size of
  the globe. All the rest of the entries are the same as in the landscape
  projection. You may want less surface roughness for a plausible look,
  unless you prefer small worlds with big topography, a la "The Little
  Prince."

  WARNING: When the "construction" process begins at the edge of the globe
  (default) or behind it, it's plotting points that will be hidden by
  subsequent points as the process sweeps around the sphere toward you.
  Our nifty hidden-point algorithms "know" this, and the first few dozen
  lines may be invisible unless a high mountain happens to poke over the
  horizon.  If you start a spherical projection and the screen stays
  black, wait for a while (a longer while for higher resolution or fill
  type 6) to see if points start to appear. Would we lie to you? If you're
  still waiting hours later, first check that the power's still on, then
  consider a faster system.


 4.8 3D Overlay Mode

  While the <3> command (see "3D" Images (p. 85)) creates its image on a
  blank screen, the <#> (or <shift-3> on some keyboards) command draws a
  second image over an existing displayed image. This image can be any
  restored image from a <R> command or the result of a just executed <3>
  command. So you can do a landscape, then press <#> and choose spherical
  projection to re-plot that image or another as a moon in the sky above

                     Fractint Version xx.xx                     Page 94

  the landscape. <#> can be repeated as many times as you like.

  It's worth noting that not all that many years ago, one of us watched
  Benoit Mandelbrot and fractal-graphics wizard Dick Voss creating just
  such a moon-over-landscape image at IBM's research center in Yorktown
  Heights, NY. The system was a large and impressive mainframe with
  floating-point facilities bigger than the average minicomputer, running
  LBLGRAPH -- what Mandelbrot calls "an independent-minded and often very
  ill-mannered heap of graphics programs that originated in work by Alex
  Hurwitz and Jack Wright of IBM Los Angeles."

  We'd like to salute LBLGRAPH, its successors, and their creators,
  because it was their graphic output (like "Planetrise over Labelgraph
  Hill," plate C9 in Mandelbrot's "Fractal Geometry of Nature") that
  helped turn fractal geometry from a mathematical curiosity into a
  phenomenon. We'd also like to point out that it wasn't as fast, flexible
  or pretty as Fractint on a 386/16 PC with S-VGA graphics. Now, a lot of
  the difference has to do with the incredible progress of micro-processor
  power since then, so a lot of the credit should go to Intel rather than
  to our highly tuned code. OK, twist our arms -- it IS awfully good code.


 4.9 Special Note for CGA or Hercules Users

  If you are one of those unfortunates with a CGA or Hercules 2-color
  monochrome graphics, it is now possible for you to make 3D projection
  images.

  Try the following unfortunately circuitous approach. Invoke Fractint,
  making sure you have set askvideo=yes. Use a disk-video mode to create a
  256 color fractal. You might want to edit the fractint.cfg file to make
  a disk-video mode with the same pixel dimensions as your normal video.
  Using the "3" command, enter the file name of the saved 256 color file,
  then select your 2 or 4 color mode, and answer the other 3D prompts. You
  will then see a 3D projection of the fractal. Another example of Stone
  Soup responsiveness to our fan mail!


 4.10 Making Terrains

  If you enjoy using Fractint for making landscapes, we have several new
  features for you to work with. When doing 3d transformations banding
  tends to occur because all pixels of a given height end up the same
  color. Now, colors can be randomized to make the transitions between
  different colors at different altitudes smoother. Use the new
  "RANDOMIZE= " variable to accomplish this. If your light source images
  all look like lunar landscapes since they are all monochrome and have
  very dark shadows, we now allow you to set the ambient light for
  adjusting the contrast of the final image. Use the "Ambient= " variable.
  In addition to being able to create scenes with light sources in
  monochrome, you can now do it in full color as well. Setting fullcolor=1
  will generate a Targa-24 file with a full color image which will be a
  combination of the original colors of the source image (or map file if
  you select map=something) and the amount of light which reflects off a
  given point on the surface. Since there can be 256 different colors in
  the original image and 256 levels of light, you can now generate an

                     Fractint Version xx.xx                     Page 95

  image with *lots* of colors. To convert it to a GIF if you can't view
  Targa files directly, you can use PICLAB (see Other Programs (p. 171)),
  and the following commands:

      SET PALETTE 256
      SET CREZ 8
      TLOAD yourfile.tga
      MAKEPAL
      MAP
      GSAVE yourfile.gif
      EXIT
  Using the full color option allows you to also set a haze factor with
  the "haze= " variable to make more distant objects more hazy.

  As a default, full color files also have the background set to sky blue.
  Warning, the files which are created with the full color option are very
  large, 3 bytes per pixel. So be sure to use a disk with enough space.
  The file is created using Fractint's disk-video caching, but is always
  created on real disk (expanded or extended memory is not used.) Try the
  following settings of the new variables in sequence to get a feel for
  the effect of each one:
      ;use this with any filltype
      map=topo
      randomize=3; adjusting this smooths color transitions

      ;now add this using filltype 5 or 6
      ambient=20; adjusting this changes the contrast
      filltype=6
      smoothing=2; makes the light not quite as granular as the terrain

      ;now add the following, and this is where it gets slow
      fullcolor=1; use PICLAB to reduce resulting lightfile to a GIF

      ;and finally this
      haze=20; sets the amount of haze for distant objects

  When full color is being used, the image you see on the screen will
  represent the amount of light being reflected, not the colors in the
  final image. Don't be disturbed if the colors look weird, they are an
  artifact of the process being used. The image being created in the
  lightfile won't look like the screen.

  However, if you are worried, hit ESC several times and when Fractint
  gets to the end of the current line it will abort. Your partial image
  will be there as LIGHT001.TGA or with whatever file name you selected
  with the lightname option. Convert it as described above and adjust any
  parameters you are not happy with. Its a little awkward, but we haven't
  figured out a better way yet.

                     Fractint Version xx.xx                     Page 96

 4.11 Making 3D Slides

  Bruce Goren, CIS's resident stereoscopic maven, contributed these tips
  on what to do with your 3D images (Bruce inspired and prodded us so much
  we automated much of what follows, allowing both this and actual on
  screen stereo viewing, but we included it here for reference and a brief
  tutorial.)

  "I use a Targa 32 video card and TOPAS graphic software, moving the
  viewport or imaginary camera left and right to create two separate views
  of the stationary object in x,y,z, space. The distance between the two
  views, known as the inter-ocular distance, toe-in or convergence angle,
  is critical. It makes the difference between good 3-D and headache-
  generating bad 3-D.

  "For a 3D fractal landscape, I created and photographed the left and
  right eye views as if flying by in an imaginary airplane and mounted the
  film chips for stereo viewing. To make my image, first I generated a
  plasma cloud based on a color map I calculated to resemble a geological
  survey map (available on CIS as TARGA.MAP). In the 3D reconstruction, I
  used a perspective value of 150 and shifted the camera -15 and +15 on
  the X-axis for the left and right views. All other values were left to
  the defaults.

  "The images are captured on a Matrix 3000 film recorder -- basically a
  box with a high-resolution (1400 lines) black and white TV and a 35mm
  camera (Konica FS-1) looking at the TV screen through a filter wheel.
  The Matrix 3000 can be calibrated for 8 different film types, but so far
  I have only used Kodak Ektachrome 64 daylight for slides and a few print
  films. I glass mount the film chips myself.

  "Each frame is exposed three times, once through each of the red, blue,
  and green filters to create a color image from computer video without
  the scan-lines which normally result from photographing television
  screens.  The aspect ratio of the resulting images led me to mount the
  chips using the 7-sprocket Busch-European Emde masks. The best source of
  Stereo mounting and viewing supplies I know of is an outfit called Reel
  3-D Enterprises, Inc. at P.O. Box 2368, Culver City, CA 90231, tel. 213-
  837-2368. "My platform is an IBM PC/AT crystal-swapped up to 9 MHz. The
  math co-processor runs on a separate 8-MHz accessory sub-board.  The
  system currently has 6.5 MB of RAM."


 4.12 Interfacing with Ray Tracing Programs

  (Also see "Ray Tracing Output", "Brief", and "Output File Name" in "3D
  Mode Selection" (p. 85).)

  Fractint allows you to save your 3d transforms in files which may be fed
  to a ray tracer (or to "Acrospin").  However, they are not ready to be
  traced by themselves. For one thing, no light source is included. They
  are actually meant to be included within other ray tracing files.

  Since the intent is to produce an object which may be included in a
  larger ray tracing scene, it is expected that all rotations, shifts, and
  final scaling will be done by the ray tracer. Thus, in creating the

                     Fractint Version xx.xx                     Page 97

  images, no facilities for rotations or shifting is provided. Scaling is
  provided to achieve the correct aspect ratio.

  WARNING! The files created using the RAY option can be huge. Setting
  COARSE to 40 will result in over 2000 triangles. Each triangle can
  utilize from 50 to 200 bytes each to describe, so your ray tracing files
  can rapidly approach or exceed 1Meg. Make sure you have enough disk
  space before you start.

  Each file starts with a comment identifying the version of Fractint by
  which it was created. The file ends with a comment giving the number of
  triangles in the file.

  The files consist of long strips of adjacent triangles. Triangles are
  clockwise or counter clockwise depending on the target ray tracer.
  Currently, MTV and Rayshade are the only ones which use counter
  clockwise triangles. The size of the triangles is set by the COARSE
  setting in the main 3d menu. Color information about each individual
  triangle is included for all files unless in the brief mode.

  To keep the poor ray tracer from working too hard, if WATERLINE is set
  to a non zero value, no triangle which lies entirely at or below the
  current setting of WATERLINE is written to the ray tracing file.  These
  may be replaced by a simple plane in the syntax of the ray tracer you
  are using.

  Fractint's coordinate system has the origin of the x-y plane at the
  upper left hand corner of the screen, with positive x to the right and
  positive y down. The ray tracing files have the origin of the x-y plane
  moved to the center of the screen with positive x to the right and
  positive y up.  Increasing values of the color index are out of the
  screen and in the +z direction. The color index 0 will be found in the
  xy plane at z=-1.

  When x- y- and zscale are set to 100, the surface created by the
  triangles will fall within a box of +/- 1.0 in all 3 directions.
  Changing scale will change the size and/or aspect ratio of the enclosed
  object.

  We will only describe the structure of the RAW format here. If you want
  to understand any of the ray tracing file formats besides RAW, please
  see your favorite ray tracer docs.

  The RAW format simply consists of a series of clockwise triangles. If
  BRIEF=yes, Each line is a vertex with coordinates x, y, and z. Each
  triangle is separated by a couple of CR's from the next. If BRIEF=no,
  the first line in each triangle description if the r,g,b value of the
  triangle.

  Setting BRIEF=yes produces shorter files with the color of each triangle
  removed - all triangles will be the same color. These files are
  otherwise identical to normal files but will run faster than the non
  BRIEF files.  Also, with BRIEF=yes, you may be able to get files with
  more triangles to run than with BRIEF=no.

                     Fractint Version xx.xx                     Page 98

  The DKB format is now obsolete. POV-Ray users should use the RAW output
  and convert to POV-Ray using the POV Group's RAW2POV utility. POV-Ray
  users can also do all 3D transformations within POV-Ray using height
  fields.



                     Fractint Version xx.xx                     Page 99

 5. Command Line Parameters, Parameter Files, Batch Mode

  Fractint accepts command-line parameters that allow you to start it with
  a particular video mode, fractal type, starting coordinates, and just
  about every other parameter and option.

  These parameters can also be specified in a SSTOOLS.INI file, to set
  them every time you run Fractint.

  They can also be specified as named groups in a .PAR (parameter) file
  which you can then call up while running Fractint by using the <@>
  command.

  In all three cases (DOS command line, SSTOOLS.INI, and parameter file)
  the parameters use the same syntax, usually a series of keyword=value
  commands like SOUND=OFF.  Each parameter is described in detail in
  subsequent sections.


 5.1 Using the DOS Command Line

  You can specify parameters when you start Fractint from DOS by using a
  command like:

      FRACTINT SOUND=OFF FILENAME=MYIMAGE.GIF

  The individual parameters are separated by one or more spaces (an
  parameter itself may not include spaces). Upper or lower case may be
  used, and parameters can be in any order.

  Since DOS commands are limited to 128 characters, Fractint has a special
  command you can use when you have a lot of startup parameters (or have a
  set of parameters you use frequently):

      FRACTINT @MYFILE

  When @filename is specified on the command line, Fractint reads
  parameters from the specified file as if they were keyed on the command
  line.  You can create the file with a text editor, putting one
  "keyword=value" parameter on each line.


 5.2 Setting Defaults (SSTOOLS.INI File)

  Every time Fractint runs, it searches the current directory, and then
  the directories in your DOS PATH, for a file named SSTOOLS.INI.  If it
  finds this file, it begins by reading parameters from it.  This file is
  useful for setting parameters you always want, such as those defining
  your printer setup.

  SSTOOLS.INI is divided into sections belonging to particular programs.
  Each section begins with a label in brackets. Fractint looks for the
  label [fractint], and ignores any lines it finds in the file belonging
  to any other label. If an SSTOOLS.INI file looks like this:

                     Fractint Version xx.xx                     Page 100

    [fractint]
    sound=off      ; (for home use only)
    printer=hp     ; my printer is a LaserJet
    inside=0       ; using "traditional" black
    [startrek]
    warp=9.5       ; Captain, I dinna think the engines can take it!

  Fractint will use only the second, third, and fourth lines of the file.
  (Why use a convention like that when Fractint is the only program you
  know of that uses an SSTOOLS.INI file?  Because there are other programs
  (such as Lee Crocker's PICLAB) that now use the same file, and there may
  one day be other, sister programs to Fractint using that file.)


 5.3 Parameter Files and the <@> Command

  You can change parameters on-the-fly while running Fractint by using the
  <@> or <2> command and a parameter file. Parameter files contain named
  groups of parameters, looking something like this:

    quickdraw {      ; a set of parameters named quickdraw
       maxiter=150
       float=no
       }
    slowdraw {       ; another set of parameters
       maxiter=2000
       float=yes
       }

  If you use the <@> or <2> command and select a parameter file containing
  the above example, Fractint will show two choices: quickdraw and
  slowdraw. You move the cursor to highlight one of the choices and press
  <Enter> to set the parameters specified in the file by that choice.

  The default parameter file name is FRACTINT.PAR. A different file can be
  selected with the "parmfile=" option, or by using <@> or <2> and then
  hitting <F6>.

  You can create parameter files with a text editor, or for some uses, by
  using the <B> command. Parameter files can be used in a number of ways,
  some examples:

    o To save the parameters for a favorite image. Fractint can do this
      for you with the <B> command.

    o To save favorite sets of 3D transformation parameters. Fractint can
      do this for you with the <B> command.

    o To set up different sets of parameters you use occasionally. For
      instance, if you have two printers, you might want to set up a group
      of parameters describing each.

    o To save image parameters for later use in batch mode - see Batch
      Mode (p. 120).

                     Fractint Version xx.xx                     Page 101

  See "Parameter Save/Restore Commands" (p. 23) for details about the <@>
  and <B> commands.


 5.4 General Parameter Syntax

  Parameters must be separated by one or more spaces.

  Upper and lower case can be used in keywords and values.

  Anything on a line following a ; (semi-colon) is ignored, i.e. is a
  comment.

  In parameter files and SSTOOLS.INI:
    o Individual parameters can be entered on separate lines.
    o Long values can be split onto multiple lines by ending a line with a
      \ (backslash) - leading spaces on the following line are ignored,
      the information on the next line from the first non-blank character
      onward is appended to the prior line.

  Some terminology:
    KEYWORD=nnn              enter a number in place of "nnn"
    KEYWORD=[filename]       you supply filename
    KEYWORD=yes|no|whatever  choose one of "yes", "no", or "whatever"
    KEYWORD=1st[/2nd[/3rd]]  the slash-separated parameters "2nd" and
                             "3rd" are optional


 5.5 Startup Parameters

  @FILENAME
  Causes Fractint to read "filename" for parameters. When it finishes, it
  resumes reading its own command line -- i.e., "FRACTINT MAXITER=250
  @MYFILE PASSES=1" is legal. This option is only valid on the DOS command
  line, as Fractint is not clever enough to deal with multiple
  indirection.

  @FILENAME/GROUPNAME
  Like @FILENAME, but reads a named group of parameters from a parameter
  file.  See "Parameter Files and the <@> Command" (p. 100).

  TEMPDIR=[directory]
  This command allows to specify the directory where Fractint writes
  temporary files.

  WORKDIR=[directory]
  This command sets the directory where miscellaneous Fractint files get
  written, including MAKEBIG.BAT and debugging files.

  FILENAME=[name]
  Causes Fractint to read the named file, which must either have been
  saved from an earlier Fractint session or be a generic GIF file, and use
  that as the starting point, bypassing the initial information screens.
  The filetype is optional and defaults to .GIF. Non-Fractint GIF files
  are restored as fractal type "plasma".
  On the DOS command line you may omit FILENAME= and just give the file

                     Fractint Version xx.xx                     Page 102

  name.

  CURDIR=yes
  Fractint uses directories set by various commands, possibly in the
  SSTOOLS.INI file. Uf you want to try out some files in the current
  directory, such as a modified copy of FRACTINT.FRM, you won't Fractint
  to read the copy in your official FRM directory. Setting curdir=yes at
  the command line will cause Fractint to look in the current directory
  for requested files first before looking in the default directory set by
  the other commands. Warning: <tab> screen may not reflect actual file
  opened in cases where the file was opened in the DOS current directory.

  BATCH=yes
  See Batch Mode (p. 120).

  AUTOKEY=play|record
  Specifying "play" runs Fractint in playback mode - keystrokes are read
  from the autokey file (see next parameter) and interpreted as if they're
  being entered from the keyboard.
  Specifying "record" runs in recording mode - all keystrokes are recorded
  in the autokey file.
  See also Autokey Mode (p. 72).

  AUTOKEYNAME=[filename]
  Specifies the file name to be used in autokey mode. The default file
  name is AUTO.KEY.

  FPU=387|IIT|NOIIT
  This parameter is useful if you have an unusual coprocessor chip. If you
  have a 80287 replacement chip with full 80387 functionality use
  "FPU=387" to inform Fractint to take advantage of those extra 387
  instructions.  If you have the IIT fpu, but don't have IIT's
  'f4x4int.com' TSR loaded, use "FPU=IIT" to force Fractint to use that
  chip's matrix multiplication routine automatically to speed up 3-D
  transformations (if you have an IIT fpu and have that TSR loaded,
  Fractint will auto-detect the presence of the fpu and TSR and use its
  extra capabilities automatically).  Since all IIT chips support 80387
  instructions, enabling the IIT code also enables Fractint's use of all
  387 instructions.  Setting "FPU=NOIIT" disables Fractint's IIT Auto-
  detect capability.  Warning: multi-tasking operating systems such as
  Windows and DesQView don't automatically save the IIT chip extra
  registers, so running two programs at once that both use the IIT's
  matrix multiply feature but don't use the handshaking provided by that
  'f4x4int.com' program, errors will result.

  MAKEDOC[=filename]
  Create Fractint documentation file (for printing or viewing with a text
  editor) and then return to DOS.  Filename defaults to FRACTINT.DOC.
  There's also a function in Fractint's online help which can be used to
  produce the documentation file - use "Printing Fractint Documentation"
  from the main help index.

  MAXHISTORY=<nnn>
  Fractint maintains a list of parameters of the past 10 images that you
  generated in the current Fractint session. You can revisit these images
  using the <h> and <Ctrl-h> commands. The maxhistory command allows you

                     Fractint Version xx.xx                     Page 103

  to set the number of image parameter sets stored in memory. The tradeoff
  is between the convenience of storing more images and memory use. Each
  image in the circular history buffer takes up over 1200 bytes, so the
  default value of ten images uses up 12,000 bytes of memory. If your
  memory is very tight, and some memory-intensive Fractint operations are
  giving "out of memory" messages, you can reduce maxistory to 2 or even
  zero. Keep in mind that every time you color cycle or change from
  integer to float or back, another image parameter set is saved, so the
  default ten images are used up quickly.


 5.6 Calculation Mode Parameters

  PASSES=1|2|3|guess|btm|tesseral
  Selects single-pass, dual-pass, triple-pass, solid-Guessing mode,
  Boundary Tracing, or the Tesseral algorithm.  See Drawing Method
  (p. 71).

  FILLCOLOR=normal|<nnn>
  Sets a color to be used for block fill by Boundary Tracing and Tesseral
  algorithms.  See Drawing Method (p. 71).

  FLOAT=yes
  Most fractal types have both a fast integer math and a floating point
  version. The faster, but possibly less accurate, integer version is the
  default. If you have a new 80486 or other fast machine with a math
  coprocessor, or if you are using the continuous potential option (which
  looks best with high bailout values not possible with our integer math
  implementation), you may prefer to use floating point. Just add
  "float=yes" to the command line to do so.  Also see "Limitations of
  Integer Math (And How We Cope)" (p. 134).

  SYMMETRY=xxx
  Forces symmetry to None, Xaxis, Yaxis, XYaxis, Origin, or Pi symmetry.
  Useful as a speedup for symmetrical fractals. This is not a kaleidoscope
  feature for imposing symmetry where it doesn't exist. Use only when the
  fractal actual exhibits the symmetry, or else results may not be
  satisfactory.


 5.7 Fractal Type Parameters

  TYPE=[name]
  Selects the fractal type to calculate. The default is type "mandel".

  PARAMS=n/n/n/n...
  Set optional (required, for some fractal types) values used in the
  calculations. These numbers typically represent the real and imaginary
  portions of some startup value, and are described in detail as needed in
  Fractal Types (p. 33).
  (Example: FRACTINT TYPE=julia PARAMS=-0.48/0.626 would wait at the
  opening screen for you to select a video mode, but then proceed straight
  to the Julia set for the stated x (real) and y (imaginary) coordinates.)

                     Fractint Version xx.xx                     Page 104

  FUNCTION=[fn1[/fn2[/fn3[/fn4]]]]
  Allows setting variable functions found in some fractal type formulae.
  Possible values are sin, cos, tan, cotan, sinh, cosh, tanh, cotanh, exp,
  log, sqr, recip (i.e. 1/z), ident (i.e. identity), cosxx (cos with a pre
  version 16 bug), asin, asinh, acos, acosh, atan, atanh, sqrt,
  abs (abs(x)+i*abs(y)), cabs (sqrt(x*x + y*y)).

  FORMULANAME=[formulaname]
  Specifies the default formula name for type=formula fractals.  (I.e. the
  name of a formula defined in the FORMULAFILE.) Required if you want to
  generate one of these fractal types in batch mode, as this is the only
  way to specify a formula name in that case.

  LNAME=[lsystemname]
  Specifies the default L-System name. (I.e. the name of an entry in the
  LFILE.) Required if you want to generate one of these fractal types in
  batch mode, as this is the only way to specify an L-System name in that
  case.

  IFS=[ifsname]
  Specifies the default IFS name. (I.e. the name of an entry in the
  IFSFILE.) Required if you want to generate one of these fractal types in
  batch mode, as this is the only way to specify an IFS name in that case.


 5.8 Image Calculation Parameters

  MAXITER=nnn
  Reset the iteration maximum (the number of iterations at which the
  program gives up and says 'OK, this point seems to be part of the set in
  question and should be colored [insidecolor]') from the default 150.
  Values range from 2 to 2,147,483,647 (super-high iteration limits like
  200000000 are useful when using logarithmic palettes).  See The
  Mandelbrot Set (p. 33) for a description of the iteration method of
  calculating fractals.
  "maxiter=" can also be used to adjust the number of orbits plotted for
  3D "attractor" fractal types such as lorenz3d and kamtorus.

  CORNERS=[xmin/xmax/ymin/ymax[/x3rd/y3rd]]
  Example: corners=-0.739/-0.736/0.288/0.291
  Begin with these coordinates as the range of x and y coordinates, rather
  than the default values of (for type=mandel) -2.0/2.0/-1.5/1.5. When you
  specify four values (the usual case), this defines a rectangle: x-
  coordinates are mapped to the screen, left to right, from xmin to xmax,
  y-coordinates are mapped to the screen, bottom to top, from ymin to
  ymax.  Six parameters can be used to describe any rotated or stretched
  parallelogram:  (xmin,ymax) are the coordinates used for the top-left
  corner of the screen, (xmax,ymin) for the bottom-right corner, and
  (x3rd,y3rd) for the bottom-left.  Entering just "CORNERS=" tells
  Fractint to use this form (the default mode) rather than CENTER-MAG (see
  below) when saving parameters with the <B> command.

  CENTER-MAG=[Xctr/Yctr/Mag[/Xmagfactor/Rotation/Skew]]
  This is an alternative way to enter corners as a center point and a
  magnification that is popular with some fractal programs and
  publications.  Entering just "CENTER-MAG=" tells Fractint to use this

                     Fractint Version xx.xx                     Page 105

  form rather than CORNERS (see above) when saving parameters with the <B>
  command.  The <TAB> status display shows the "corners" in both forms.
  When you specify three values (the usual case), this defines a
  rectangle:  (Xctr, Yctr) specifies the coordinates of the center of the
  image while Mag indicates the amount of magnification to use.  Six
  parameters can be used to describe any rotated or stretched
  parallelogram:  Xmagfactor tells how many times bigger the x-
  magnification is than the y-magnification, Rotation indicates how many
  degrees the image has been turned, and Skew tells how many degrees the
  image is leaning over.  Positive angles will rotate and skew the image
  counter-clockwise.

  BAILOUT=nnn
  Over-rides the default bailout criterion for escape-time fractals. Can
  also be set from the parameters screen after selecting a fractal type.
  See description of bailout in The Mandelbrot Set (p. 33).

  BAILOUTEST=mod|real|imag|or|and
  Specifies the Bailout Test (p. 81) used to determine when the fractal
  calculation has exceeded the bailout value.  The default is mod and not
  all fractal types can utilize the additional tests.

  RESET
  Causes Fractint to reset all calculation related parameters to their
  default values. Non-calculation parameters such as "printer=", "sound=",
  and "savename=" are not affected. RESET should be specified at the start
  of each parameter file entry (used with the <@> command) which defines
  an image, so that the entry need not describe every possible parameter -
  when invoked, all parameters not specifically set by the entry will have
  predictable values (the defaults).

  INITORBIT=pixel
  INITORBIT=nnn/nnn
  Allows control over the value used to begin each Mandelbrot-type orbit.
  "initorbit=pixel" is the default for most types; this command
  initializes the orbit to the complex number corresponding to the screen
  pixel. The command "initorbit=nnn/nnn" uses the entered value as the
  initializer. See the discussion of the Mandellambda Sets (p. 39) for
  more on this topic.

  ORBITDELAY=<nn>
  Slows up the display of orbits using the <o> command for folks with hot
  new computers. Units are in 1/10000 seconds per orbit point.
  ORBITDELAY=10 therefore allows you to see each pixel's orbit point for
  about one millisecond. For best display of orbits, try passes=1 and a
  moderate resolution such as 320x200.  Note that the first time you press
  the 'o' key with the 'orbitdelay' function active, your computer will
  pause for a half-second or so to calibrate a high-resolution timer.

  SHOWORBIT=yes|no
  Causes the during-generation orbits feature toggled by the <O> command
  to start off in the "on" position each time a new fractal calculation
  starts.

                     Fractint Version xx.xx                     Page 106

  PERIODICITY=no|show|nnn
  Controls periodicity checking (see Periodicity Logic (p. 134)).  "no"
  turns it off, "show" lets you see which pixels were painted as "inside"
  due to being caught by periodicity.  Specifying a number causes a more
  conservative periodicity test (each increase of 1 divides test tolerance
  by 2).  Entering a negative number lets you turn on "show" with that
  number. Type lambdafn function=exp needs periodicity turned off to be
  accurate -- there may be other cases.

  RSEED=nnnn
  The initial random-number "seed" for plasma clouds is taken from your
  PC's internal clock-timer. This argument forces a value (which you can
  see in the <Tab> display), and allows you to reproduce plasma clouds. A
  detailed discussion of why a TRULY random number may be impossible to
  define, let alone generate, will have to wait for "FRACTINT: The 3-MB
  Doc File."

  SHOWDOT=<nn>
  Colors the pixel being calculated color <nn>. Useful for very slow
  fractals for showing you the calculation status.

  ASPECTDRIFT=<nn>
  When zooming in or out, the aspect ratio (the width to height ratio) can
  change slightly due to rounding and the noncontinuous nature of pixels.
  If the aspect changes by a factor less than <nn>, then the aspect is set
  to it's normal value, making the center-mag Xmagfactor parameter equal
  to 1.  (see CENTER-MAG above.)  The default is 0.01.  A larger value
  adjusts more often.  A value of 0 does no adjustment at all.


 5.9 Color Parameters

  INSIDE=nnn|bof60|bof61|zmag|attractor|epscross|startrail|period
  Set the color of the interior: for example, "inside=0" makes the M-set
  "lake" a stylish basic black. A setting of -1 makes inside=maxiter.

  Four more options reveal hidden structure inside the lake.  Inside=bof60
  and inside=bof61, are named after the figures on pages 60 and 61 of
  "Beauty of Fractals".  Inside=zmag is a method of coloring based on the
  magnitude of Z after the maximum iterations have been reached.  The
  affect along the edges of the Mandelbrot is like thin-metal welded
  sculpture.  Inside=period colors pixels according to the period of their
  eventual orbit.  See Inside=bof60|bof61|zmag|period (p. 151) for a
  brilliant explanation of what these do!

  Inside=epscross colors pixels green or yellow according to whether their
  orbits swing close to the Y-axis or X-axis, respectively.
  Inside=starcross has a coloring scheme based on clusters of points in
  the orbits. Best with outside=<nnn>. For more information, see
  Inside=epscross|startrail (p. 151).

  Note that the "Look for finite attractor" option on the <Y> options
  screen will override the selected inside option if an attractor is found
  - see Finite Attractors (p. 152).

                     Fractint Version xx.xx                     Page 107

  OUTSIDE=nnn|iter|real|imag|summ|mult|atan
  The classic method of coloring outside the fractal is to color according
  to how many iterations were required before Z reached the bailout value,
  usually 4. This is the method used when OUTSIDE=iter.

  However, when Z reaches bailout the real and imaginary components can be
  at very diferent values.  OUTSIDE=real and OUTSIDE=imag color using the
  iteration value plus the real or imaginary values.  OUTSIDE=summ uses
  the sum of all these values.  These options can give a startling 3d
  quality to otherwise flat images and can change some boring images to
  wonderful ones. OUTSIDE=mult colors by multiplying the iteration by real
  divided by imaginary. There was no mathematical reason for this, it just
  seemed like a good idea.  OUTSIDE=atan colors by determining the angle
  in degrees the last iterated value has with respect to the real axis,
  and using the absolute value.

  Outside=nnn sets the color of the exterior to some number of your
  choosing: for example, "OUTSIDE=1" makes all points not INSIDE the
  fractal set to color 1 (blue). Note that defining an OUTSIDE color
  forces any image to be a two-color one: either a point is INSIDE the
  set, or it's OUTSIDE it.

  MAP=[filename]
  Reads in a replacement color map from [filename]. This map replaces the
  default color map of your video adapter. Requires a VGA or higher
  adapter.  The difference between this argument and an alternate map read
  in via <L> in color-command mode is that this one applies to the entire
  run.  See Palette Maps (p. 72).

  COLORS=@filename|colorspecification
  Sets colors for the current image, like the <L> function in color
  cycling and palette editing modes. Unlike the MAP= parameter, colors set
  with COLORS= do not replace the default - when you next select a new
  fractal type, colors will revert to their defaults.
  COLORS=@filename tells Fractint to use a color map file named
  "filename".  See Palette Maps (p. 72).
  COLORS=colorspecification specifies the colors directly. The value of
  "colorspecification" is rather long (768 characters for 256 color
  modes), and its syntax is not documented here.  This form of the COLORS=
  command is not intended for manual use - it exists for use by the <B>
  command when saving the description of a nice image.

  CYCLERANGE=nnn/nnn
  Sets the range of color numbers to be animated during color cycling.
  The default is 1/255, i.e. just color number 0 (usually black) is not
  cycled.

  CYCLELIMIT=nnn
  Sets the speed of color cycling. Technically, the number of DAC
  registers updated during a single vertical refresh cycle. Legal values
  are 1 - 256, default is 55.

  TEXTCOLORS=mono
  Set text screen colors to simple black and white.

                     Fractint Version xx.xx                     Page 108

  TEXTCOLORS=aa/bb/cc/...
  Set text screen colors. Omit any value to use the default (e.g.
  textcolors=////50 to set just the 5th value). Each value is a 2 digit
  hexadecimal value; 1st digit is background color (from 0 to 7), 2nd
  digit is foreground color (from 0 to F).
  Color values are:
      0 black     8 gray
      1 blue      9 light blue
      2 green     A light green
      3 cyan      B light cyan
      4 red       C light red
      5 magenta   D light magenta
      6 brown     E yellow
      7 white     F bright white
  31 colors can be specified, their meanings are as follows:
    heading:
      1  Fractint version info
      2  heading line development info (not used in released version)
    help:
      3  sub-heading
      4  main text
      5  instructions at bottom of screen
      6  hotlink field
      7  highlighted (current) hotlink
    menu, selection boxes, parameter input boxes:
      8  background around box and instructions at bottom
      9  emphasized text outside box
     10  low intensity information in box
     11  medium intensity information in box
     12  high intensity information in box (e.g. heading)
     13  current keyin field
     14  current keyin field when it is limited to one of n values
     15  current choice in multiple choice list
     16  speed key prompt in multiple choice list
     17  speed key keyin in multiple choice list
    general (tab key display, IFS parameters, "thinking" display):
     18  high intensity information
     19  medium intensity information
     20  low intensity information
     21  current keyin field
    disk video:
     22  background around box
     23  high intensity information
     24  low intensity information
    diagnostic messages:
     25  error
     26  information
    credits screen:
     27  bottom lines
     28  high intensity divider line
     29  low intensity divider line
     30  primary authors
     31  contributing authors
  The default is
     textcolors=1F/1A/2E/70/28/71/31/78/70/17/1F/1E/2F/3F/5F/07/
                0D/71/70/78/0F/70/0E/0F/4F/20/17/20/28/0F/07

                     Fractint Version xx.xx                     Page 109

  (In a real command file, all values must be on one line.)
  OLDDEMMCOLORS=yes|no
  Sets the coloring scheme used with the distance estimator method to the
  pre-version 16 scheme.


 5.10 Doodad Parameters

  LOGMAP=yes|old|n
  Selects a compressed relationship between escape-time iterations and
  palette colors.  See "Logarithmic Palettes and Color Ranges" (p. 77)
  for details.

  RANGES=nn/nn/nn/...
  Specifies ranges of escape-time iteration counts to be mapped to each
  color number.  See "Logarithmic Palettes and Color Ranges" (p. 77) for
  details.

  DISTEST=nnn/nnn
  A nonzero value in the first parameter enables the distance estimator
  method. The second parameter specifies the "width factor", defaults to
  71.  See "Distance Estimator Method" (p. 74) for details.

  DECOMP=2|4|8|16|32|64|128|256
  Invokes the corresponding decomposition coloring scheme.  See
  Decomposition (p. 76) for details.

  BIOMORPH=nnn
  Turn on biomorph option; set affected pixels to color nnn.  See
  Biomorphs (p. 78) for details.

  POTENTIAL=maxcolor[/slope[/modulus[/16bit]]]
  Enables the "continuous potential" coloring mode for all fractal types
  except plasma clouds, attractor types such as lorenz, and IFS. The four
  arguments define the maximum color value, the slope of the potential
  curve, the modulus "bailout" value, and whether 16 bit values are to be
  calculated.  Example: "POTENTIAL=240/2000/40/16bit". The Mandelbrot and
  Julia types ignore the modulus bailout value and use their own hardwired
  value of 4.0 instead.  See Continuous Potential (p. 79) for details.

  INVERT=nn/nn/nn
  Turns on inversion. The parameters are radius of inversion, x-coordinate
  of center, and y-coordinate of center. -1 as the first parameter sets
  the radius to 1/6 the smaller screen dimension; no x/y parameters
  defaults to center of screen. The values are displayed with the <Tab>
  command.  See Inversion (p. 76) for details.

  FINATTRACT=no|yes
  Another option to show coloring inside some Julia "lakes" to show escape
  time to finite attractors. Works with lambda, magnet types, and possibly
  others.  See Finite Attractors (p. 152) for more information.

  EXITNOASK=yes
  This option forces Fractint to bypass the final "are you sure?" exit
  screen when the ESCAPE key is pressed from the main image-generation
  screen.  Added at the request of Ward Christensen.  It's his funeral

                     Fractint Version xx.xx                     Page 110

  <grin>.


 5.11 File Parameters

  In Fractint you can use various filename variables to specify files, set
  default directories, or both. For example, in the SAVENAME description
  below, [name] can be a filename, a directory name, or a fully qualified
  pathname plus filename. You can specify default directories using these
  variables in your SSTOOLS.INI file.

  SAVENAME=[name]
  Set the filename to use when you <S>ave a screen. The default filename
  is FRACT001. The .GIF extension is optional (Example: SAVENAME=myfile)

  OVERWRITE=no|yes
  Sets the savename overwrite flag (default is 'no'). If 'yes', saved
  files will over-write existing files from previous sessions; otherwise
  the automatic incrementing of FRACTnnn.GIF will find the first unused
  filename.

  SAVETIME=nnn
  Tells Fractint to automatically do a save every nnn minutes while a
  calculation is in progress.  This is mainly useful with long batches -
  see Batch Mode (p. 120).
  GIF87a=YES
  Backward-compatibility switch to force creation of GIF files in the
  GIF87a format. As of version 14, Fractint defaults to the new GIF89a
  format which permits storage of fractal information within the format.
  GIF87a=YES is only needed if you wish to view Fractint images with a GIF
  decoder that cannot accept the newer format.  See GIF Save File Format
  (p. 166).

  DITHER=YES
  Dither a color file into two colors for display on a b/w display.  This
  give a poor-quality display of gray levels.  Note that if you have a 2-
  color display, you can create a 256-color gif with disk video and then
  read it back in dithered.

  PARMFILE=[parmfilename]
  Specifies the default parameter file to be used by the <@> (or <2>) and
  <B> commands.  If not specified, the default is FRACTINT.PAR.

  FORMULAFILE=[formulafilename]
  Specifies the formula file for type=formula fractals (default is
  FRACTINT.FRM).  Handy if you want to generate one of these fractal types
  in batch mode.

  LFILE=[lsystemfile]
  Specifies the default L-System file for type=lsystem fractals (if not
  FRACTINT.L).

  IFSFILE=[ifsfilename]
  Specifies the default file for type=ifs fractals (default is
  FRACTINT.IFS).

                     Fractint Version xx.xx                     Page 111

  FILENAME=[.suffix]
  Sets the default file extension used for the <r> command.  When this
  parameter is omitted, the default file mask shows .GIF and .POT files.
  You might want to specify this parameter and the SAVENAME= parameter in
  your SSTOOLS.INI file if you keep your fractal images separate from
  other .GIF files by using a different suffix for them.

  ORBITSAVE=yes
  Causes the file ORBITS.RAW to be opened and the points generated by
  orbit fractals or IFS fractals to be saved in a raw format. This file
  can be read by the Acrospin program which can rotate and scale the image
  rapidly in response to cursor-key commands. The filename ORBITS.RAW is
  fixed and will be overwritten each time a new fractal is generated with
  this option.
  (see Barnsley IFS Fractals (p. 43) Orbit Fractals (p. 50) Acrospin
  (p. 171));


 5.12 Video Parameters

  VIDEO=xxx
  Set the initial video mode (and bypass the informational screens). Handy
  for batch runs. (Example: VIDEO=F4 for IBM 16-color VGA.)  You can
  obtain the current VIDEO= values (key assignments) from the "select
  video mode" screens inside Fractint. If you want to do a batch run with
  a video mode which isn't currently assigned to a key, you'll have to
  modify the key assignments - see "Video Mode Function Keys" (p. 28).

  ASKVIDEO=yes|no
  If "no," this eliminates the prompt asking you if a file to be restored
  is OK for your current video hardware.
  WARNING: every version of Fractint so far has had a bigger, better, but
  shuffled-around video table. Since calling for a mode your hardware
  doesn't support can leave your system in limbo, be careful about leaving
  the above two parameters in a command file to be used with future
  versions of Fractint, particularly for the super-VGA modes.


  ADAPTER=hgc|cga|ega|egamono|mcga|vga|ATI|Everex|Trident|NCR|Video7|Genoa|
          Paradise|Chipstech|Tseng3000|Tseng4000|AheadA|AheadB|Oaktech
  Bypasses Fractint's internal video autodetect logic and assumes that the
  specified kind of adapter is present. Use this parameter only if you
  encounter video problems without it.  Specifying adapter=vga with an
  SVGA adapter will make its extended modes unusable with Fractint.  All
  of the options after the "VGA" option specify specific SuperVGA chipsets
  which are capable of video resolutions higher than that of a "vanilla"
  VGA adapter.  Note that Fractint cares about the Chipset your adapter
  uses internally, not the name of the company that sold it to you.

  VESADETECT=yes|no
  Specify no to bypass VESA video detection logic. Try this if you
  encounter video problems with a VESA compliant video adapter or driver.

  AFI=yes|8514|no
  Normally, when you attempt to use an 8514/A-specific video mode,
  Fractint first attempts to detect the presence of an 8514/A register-

                     Fractint Version xx.xx                     Page 112

  compatible adapter.  If it fails to find one, it then attempts to detect
  the presence of an 8514/A-compatible API (IE, IBM's HDILOAD or its
  equivalent).  Fractint then uses either its register-compatible or its
  API-compatible video logic based on the results of those tests.  If you
  have an "8514/A-compatible" video adapter that passes Fractint's
  register-compatible detection logic but doesn't work correctly with
  Fractint's register-compatible video logic, setting "afi=yes" will force
  Fractint to bypass the register-compatible code and look only for the
  API interface.

  TEXTSAFE=yes|no|bios|save
  When you switch from a graphics image to text mode (e.g. when you use
  <F1> while a fractal is on display), Fractint remembers the graphics
  image, and restores it when you return from the text mode.  This should
  be no big deal - there are a number of well-defined ways Fractint could
  do this which *should* work on any video adapter.  They don't - every
  fast approach we've tried runs into a bug on one video adapter or
  another.  So, we've implemented a fast way which works on most adapters
  in most modes as the default, and added this parameter for use when the
  default approach doesn't work.
  If you experience the following problems, please fool around with this
  parameter to try to fix the problem:
    o Garbled image, or lines or dashes on image, when returning to image
      after going to menu, <tab> display, or help.
    o Blank screen when starting Fractint.
  The problems most often occur in higher resolution modes. We have not
  encountered them at all in modes under 320x200x256 - for those modes
  Fractint always uses a fast image save/restore approach.
  Textsafe options:
    yes: This is the default. When switching to/from graphics, Fractint
      saves just that part of video memory which EGA/VGA adapters are
      supposed to modify during the mode changes.
    no: This forces use of monochrome 640x200x2 mode for text displays
      (when there is a high resolution graphics image to be saved.) This
      choice is fast but uses chunky and colorless characters. If it turns
      out to be the best choice for you, you might want to also specify
      "textcolors=mono" for a more consistent appearance in text screens.
    bios: This saves memory in the same way as textsafe=yes, but uses the
      adapter's BIOS routines to save/restore the graphics state.  This
      approach is fast and ought to work on all adapters. Sadly, we've
      found that very few adapters implement this function perfectly.
    save: This is the last choice to try. It should work on all adapters
      in all modes but it is slow. It tells Fractint to save/restore the
      entire image. Expanded or extended memory is used for the save if
      you have enough available; otherwise a temporary disk file is used.
      The speed of textsafe=save will be acceptable on some machines but
      not others.  The speed depends on:
        o Cpu and video adapter speed.
        o Whether enough expanded or extended memory is available.
        o Video mode of image being remembered. A few special modes are
          *very* slow compared to the rest. The slow ones are: 2 and 4 color
          modes with resolution higher than 640x480; custom modes for ATI
          EGA Wonder, Paradise EGA-480, STB, Compaq portable 386, AT&T 6300,
          and roll-your-own video modes implemented with customized
          "yourvid.c" code.
  If you want to tune Fractint to use different "textsafe" options for

                     Fractint Version xx.xx                     Page 113

  different video modes, see "Customized Video Modes, FRACTINT.CFG"
  (p. 126).  (E.g. you might want to use the slower textsafe=save approach
  just for a few high-resolution modes which have problems with
  textsafe=yes.)

  EXITMODE=nn
  Sets the bios-supported videomode to use upon exit to the specified
  value.  nn is in hexadecimal.  The default is 3, which resets to 80x25
  color text mode on exit. With Hercules Graphics Cards, and with
  monochrome EGA systems, the exit mode is always 7 and is unaffected by
  this parameter.

  TPLUS=yes|no
  For TARGA+ adapters. Setting this to 'no' pretends a TARGA+ is NOT
  installed.

  NONINTERLACED=yes|no
  For TARGA+ adapters. Setting this to 'yes' will configure the adapter to
  a non-interlaced mode whenever possible.  It should only be used with a
  multisynch monitor. The default is no, i.e. interlaced.

  MAXCOLORRES=8|16|24
  For TARGA+ adapters. This determines the number of bits to use for color
  resolution.  8 bit color is equivalent to VGA color resolution. The 16
  and 24 bit color resolutions are true color video modes which are not
  yet supported by Fractint but are hopefully coming soon.

  PIXELZOOM=0|1|2|3
  For TARGA+ adapters. Lowers the video mode resolution by powers of 2.
  For example, the 320x200 video resolution on the TARGA+ is actually the
  640x400 video mode with a pixel zoom of 1.  Using the 640x400 video mode
  with a zoom of 3 would lower the resolution by 8, which is 2 raised to
  the 3rd power, for a full screen resolution of 80x50 pixels.

  VIEWWINDOWS=xx[/xx[/yes|no[/nn[/nn]]]] Set the reduction factor, final
  media aspect ratio, crop starting coordinates (y/n), explicit x size,
  and explicit y size, see "View Window" (p. 26).


 5.13 Sound Parameters

  SOUND=off|x|y|z
  We're all MUCH too busy to waste time with Fractint at work, and no
  doubt you are too, so "sound=off" is included only for use at home, to
  avoid waking the kids or your Significant Other, late at night. (By the
  way, didn't you tell yourself "just one more zoom on LambdaSine" an hour
  ago?)  Suggestions for a "boss" hot-key will be cheerfully ignored, as
  this sucker is getting big enough without including a spreadsheet screen
  too.  The "sound=x/y/x" options are for the "attractor" fractals, like
  the Lorenz fractals - they play with the sound on your PC speaker as
  they are generating an image, based on the X or Y or Z co-ordinate they
  are displaying at the moment.  At the moment, "sound=x" (or y or z)
  really doesn't work very well when using an integer algorithm - try it
  with the floating-point toggle set, instead.

                     Fractint Version xx.xx                     Page 114

  The scope of the sound command has been extended. You can now hear the
  sound of fractal orbits--just turn on sound from the command line or the
  <X> menu, fire up a fractal, and try the <O>rbits command. Use the
  orbitdelay=<nnn> command (also on the <X> menu) to dramatically alter
  the effect, which ranges from an unearthly scream to a series of
  discrete tones. Not recommended when people you have to live with are
  nearby!  Remember, we don't promise that it will sound beautiful!

  You can also "hear" any image that Fractint can decode; turn on sound
  before using <R> to read in a GIF file. We have no idea if this feature
  is useful. It was inspired by the comments of an on-line friend who is
  blind. We solicit feedback and suggestions from anyone who finds these
  sound features interesting or useful. The orbitdelay command also
  affects the sound of decoding images.

  HERTZ=nnn
  Adjusts the sound produced by the "sound=x/y/z" option.  Legal values
  are 20 through 15000.


 5.14 Printer Parameters

  PRINTER=type[/resolution[/port#]]
  Defines your printer setup. The SSTOOLS.INI file is a REAL handy place
  to put this option, so that it's available whenever you have that
  sudden, irresistible urge for hard copy.
  Printer types:
    IB  IBM-compatible (default)
    EP  Epson-compatible
    HP  LaserJet
    CO  Star Micronics Color printer, supposedly Epson-color-compatible
    PA  Paintjet
    PS  PostScript
    PSL Postscript, landscape mode
    PL  Plotter using HP-GL
  Resolution:
    In dots per inch.
    Epson/IBM: 60, 120, 240
    LaserJet: 75, 150, 300
    PaintJet: 90, 180
    PostScript: 10 through 600, or special value 0 to print full page to
    within about .4" of the edges (in portrait mode, width is full page and
    height is adjusted to 3:4 aspect ratio)
    Plotter: 1 to 10 for 1/Nth of page (e.g. 2 for 1/2 page)
  Port:
    1, 2, 3 for LPT1-3 via BIOS
    11, 12, 13, 14 for COM1-4 via BIOS
    21, 22 for LPT1 or LPT2 using direct port access (faster when it works)
    31, 32 for COM1 or COM2 using direct port access

  COMPORT=port/baud/options
  Serial printer port initialization.
  Port=1,2,3,etc.
  Baud=115,150,300,600,1200,2400,4800,9600
  Options: 7,8 | 1,2 | e,n,o (any order).
  Example: comport=1/9600/n81 for COM1 set to 9600, no parity, 8 bits per

                     Fractint Version xx.xx                     Page 115

  character, 1 stop bit.

  LINEFEED=crlf|lf|cr
  Specifies the control characters to emit at end of each line:  carriage
  return and linefeed, just linefeed, or just carriage return.  The
  default is crlf.

  TITLE=yes
  If specified, title information is added to printouts.

  PRINTFILE=filename
  Causes output data for the printer to be written to the named file on
  disk instead of to a printer port. The filename is incremented by 1 each
  time an image is printed - e.g. if the name is FRAC01.PRN, the second
  print operation writes to FRAC02.PRN, etc. Existing files are not
  overwritten - if the file exists, the filename is incremented to a new
  name.


 5.15 PostScript Parameters

  EPSF=1|2|3
  Forces print-to-file and PostScript. If PRINTFILE is not specified, the
  default filename is FRACT001.EPS. The number determines how 'well-
  behaved' a .EPS file is. 1 means by-the-book. 2 allows some EPS 'no-nos'
  like settransfer and setscreen - BUT includes code that should make the
  code still work without affecting the rest of the non-EPS document. 3 is
  a free-for-all.

  COLORPS=YES|NO - Enable or disable the color extensions.

  RLEPS=YES|NO
  Enable or disable run length encoding of the PostScript file.  Run
  length encoding will make the PostScript file much smaller, but it may
  take longer to print.  The run length encoding code is based on pnmtops,
  which is copyright (C) 1989 by Jef Poskanzer, and carries the following
  notice: "Permission to use, copy, modify, and distribute this software
  and its documentation for any purpose and without fee is hereby granted,
  provided that the above copyright notice appear in all copies and that
  both that copyright notice and this permission notice appear in
  supporting documentation.  This software is provided "as is" without
  express or implied warranty."

  TRANSLATE=yes|-n|n
  Translate=yes prints the negative image of the fractal.  Translate=n
  reduces the image to that many colors. A negative value causes a color
  reduction as well as a negative image.

  HALFTONE=frq/ang/sty[/f/a/s/f/a/s/f/a/s]
  Tells the PostScript printer how to define its halftone screen. The
  first value, frequency, defines the number of halftone lines per inch.
  The second chooses the angle (in degrees) that the screen lies at. The
  third option chooses the halftone 'spot' style. Good default frequencies
  are between 60 and 80; Good default angles are 45 and 0; the default
  style is 0. If the halftone= option is not specified, Fractint will
  print using the printer's default halftone screen, which should have

                     Fractint Version xx.xx                     Page 116

  been already set to do a fine job on the printer.

  These are the only three used when colorps=no. When color PS printing is
  being used, the other nine options specify the red, green, then blue
  screens. A negative number in any of these places will cause it to use
  the previous (or default) value for that parameter. NOTE: Especially
  when using color, the built-in screens in the printer's ROM may be the
  best choice for printing.
  The default values are as follows:
  halftone=45/45/1/45/75/1/45/15/1/45/0/1 and these will be used if
  Fractint's halftone is chosen over the printer's built-in screen.

  Current halftone styles:
      0 Dot
      1 Dot (Smoother)
      2 Dot (Inverted)
      3 Ring (Black)
      4 Ring (White)
      5 Triangle (Right)
      6 Triangle (Isosceles)
      7 Grid
      8 Diamond
      9 Line
     10 Microwaves
     11 Ellipse
     12 Rounded Box
     13 Custom
     14 Star
     15 Random
     16 Line (slightly different)

  A note on device-resolution black and white printing
  ----------------------------------------------------

  This mode of printing can now be done much more quickly, and takes a lot
  less file space. Just set EPSF=0 PRINTER=PSx/nnn COLORPS=NO RLEPS=YES
  TRANSLATE=m, where x is P or L for portrait/landscape, nnn is your
  printer's resolution, m is 2 or -2 for positive or negative printing
  respectively. This combination of parameters will print exactly one
  printer pixel per each image pixel and it will keep the proportions of
  the picture, if both your screen and printer have square pixels (or the
  same pixel-aspect). Choose a proper (read large) window size to fill as
  much of the paper as possible for the most spectacular results.  2048 by
  2048 is barely enough to fill the width of a letter size page with 300
  dpi printer resolution.  For higher resolution printers, you will wish
  fractint supported larger window sizes (hint, hint...). Bug reports
  and/or suggestions should be forwarded to Yavuz Onder through e-mail
  (yavuz@bnr.ca).

  A word from the author (Scott Taylor)
  -------------------------------------

  Color PostScript printing is new to me. I don't even have a color
  printer to test it on. (Don't want money. Want a Color PostScript
  printer!) The initial tests seem to have worked. I am still testing and
  don't know whether or not some sort of gamma correction will be needed.

                     Fractint Version xx.xx                     Page 117

  I'll have to wait and see about that one.


 5.16 PaintJet Parameters

  Note that the pixels printed by the PaintJet are square.  Thus, a
  printout of an image created in a video mode with a 4:3 pixel ratio
  (such as 640x480 or 800x600) will come out matching the screen; other
  modes (such as 320x200) will come out stretched.

  Black and white images, or images using the 8 high resolution PaintJet
  colors, come out very nicely.  Some images using the full spectrum of
  PaintJet colors are very nice, some are disappointing.

  When 180 dots per inch is selected (in PRINTER= command), high
  resolution 8 color printing is done.  When 90 dpi is selected, low
  resolution printing using the full 330 dithered color palette is done.
  In both cases, Fractint starts by finding the nearest color supported by
  the PaintJet for each color in your image.  The translation is then
  displayed (unless the current display mode is disk video).  This display
  *should* be a fairly good match to what will be printed - it won't be
  perfect most of the time but should give some idea of how the output
  will look.  At this point you can <Enter> to go ahead and print, <Esc>
  to cancel, or <k> to cancel and keep the adjusted colors.

  Note that you can use the color map PAINTJET.MAP to create images which
  use the 8 high resolution colors available on the PaintJet.  Also, two
  high-resolution disk video modes are available for creating full page
  images.

  If you find that the preview image seems very wrong (doesn't match what
  actually gets printed) or think that Fractint could be doing a better
  job of picking PaintJet colors to match your image's colors, you can try
  playing with the following parameter.  Fair warning: this is a very
  tricky business and you may find it a very frustrating business trying
  to get it right.

  HALFTONE=r/g/b
  (The parameter name is not appropriate - we appropriated a PostScript
  parameter for double duty here.)
  This separately sets the "gamma" adjustment for each of the red, green,
  and blue color components.  Think of "gamma" as being like the contrast
  adjustment on your screen.  Higher gamma values for all three components
  results in colors with more contrast being produced on the printer.
  Since each color component can have its gamma separately adjusted, you
  can change the resulting color mix subtly (or drastically!)
  Each gamma value entered has one implied decimal digit.
  The default is "halftone=21/19/16", for red 2.1, green 1.9, and blue
  1.6.  (A note from Pieter Branderhorst: I wrote this stuff to come out
  reasonably on my monitor/printer.  I'm a bit suspicious of the guns on
  my monitor; if the colors seem ridiculously wrong on your system you
  might start by trying halftone=17/17/17.)

                     Fractint Version xx.xx                     Page 118

 5.17 Plotter Parameters

  Plotters which understand HP-GL commands are supported. To use a
  plotter, draw a SMALL image (32x20 or 64x40) using the <v>iew screen
  options.  Put a red pen in the first holder in the plotter, green in the
  second, blue in the third.  Now press <P> to start plotting.  Now get a
  cup of coffee...  or two... or three.  It'll take a while to plot.
  Experiment with different resolutions, plot areas, plotstyles, and even
  change pens to create weird-colored images.

  PLOTSTYLE=0|1|2
  0: 3 parallel lines (red/green/blue) are drawn for each pixel, arranged
    like "///".  Each bar is scaled according to the intensity of the
    corresponding color in the pixel.  Using different pen colors (e.g.
    blue, green, violet) can come out nicely.  The trick is to not tell
    anyone what color the bars are supposed to represent and they will
    accept these plotted colors because they do look nice...
  1: Same as 0, but the lines are also twisted.  This removes some of the
    'order' of the image which is a nice effect.  It also leaves more
    whitespace making the image much lighter, but colors such as yellow
    are actually visible.
  2: Color lines are at the same angle and overlap each other.  This type
    has the most whitespace.  Quality improves as you increase the number
    of pixels squeezed into the same size on the plotter.


 5.18 3D Parameters

  To stay out of trouble, specify all the 3D parameters, even if you want
  to use what you think are the default values. It takes a little practice
  to learn what the default values really are. The best way to create a
  set of parameters is to use the <B> command on an image you like and
  then use an editor to modify the resulting parameter file.

  3D=Yes
  3D=Overlay
  Resets all 3d parameters to default values. If FILENAME= is given,
  forces a restore to be performed in 3D mode (handy when used with
  'batch=yes' for batch-mode 3D images). If specified, 3D=Yes should come
  before any other 3d parameters on the command line or in a parameter
  file entry. The form 3D=Overlay is identical except that the previous
  graphics screen is not cleared, as with the <#> (<shift-3> on some
  keyboards) overlay command.  Useful for building parameter files that
  use the 3D overlay feature.

  The options below override the 3D defaults:
  PREVIEW=yes                Turns on 3D 'preview' default mode
  SHOWBOX=yes                Turns on 3D 'showbox' default mode
  COARSE=nn                  Sets Preview 'coarseness' default value
  SPHERE=yes                 Turns on spherical projection mode
  STEREO=n                   Selects the type of stereo image creation
  RAY=nnn                    selects raytrace output file format
  BRIEF=yes                  selects brief or verbose file for DKB output
  USEGRAYSCALE=yes           use grayscale as depth instead of color number

                     Fractint Version xx.xx                     Page 119

  INTEROCULAR=nn             Sets the interocular distance for stereo
  CONVERGE=nn                Determines the overall image separation
  CROP=nn/nn/nn/nn           Trims the edges off stereo pairs
  BRIGHT=nn/nn               Compensates funny glasses filter parameters
  LONGITUDE=nn/nn            Longitude minimum and maximum
  LATITUDE=nn/nn             Latitude minimum and maximum
  RADIUS=nn                  Radius scale factor
  ROTATION=nn[/nn[/nn]]      Rotation about x,y, and z axes
  SCALEZYZ=nn/nn/nn          X,y,and z scale factors
  ROUGHNESS=nn               Same as z scale factor
  WATERLINE=nn               Colors nn and below will be "inside" color
  FILLTYPE=nn                3D filltype
  PERSPECTIVE=nn             Perspective distance
  XYSHIFT=nn/nn              Shift image in x and y directions with
                              perspective
  LIGHTSOURCE=nn/nn/nn       Coordinates for light-source vector
  SMOOTHING=nn               Smooths images in light-source fill modes
  TRANSPARENT=min/max        Defines a range of colors to be treated as
                              "transparent" when <#>Overlaying 3D images.
  XYADJUST=nn/nn             This shifts the image in the x/y dir without
                              perspective

  Below are new commands as of version 14 that support Marc Reinig's
  terrain features.

  RANDOMIZE=nnn (0 - 100)
  This feature randomly varies the color of a pixel to near by colors.
  Useful to minimize map banding in 3d transformations. Usable with all
  FILLTYPES. 0 disables, max values is 7. Try 3 - 5.

  AMBIENT=nnn (0 - 100)
  Set the depth of the shadows when using full color and light source
  filltypes. "0" disables the function, higher values lower the contrast.

  FULLCOLOR=yes
  Valid with any light source FILLTYPE. Allows you to create a Targa-24
  file which uses the color of the image being transformed or the map you
  select and shades it as you would see it in real life. Well, its better
  than B&W.  A good map file to use is topo

  HAZE=nnn (0 - 100)
  Gives more realistic terrains by setting the amount of haze for distant
  objects when using full color in light source FILLTYPES. Works only in
  the "y" direction currently, so don't use it with much y rotation. Try
  "rotation=85/0/0". 0 disables.

  LIGHTNAME=<filename>
  The name of the Targa-24 file to be created when using full color with
  light source. Default is light001.tga. If overwrite=no (the default),
  the file name will be incremented until an unused filename is found.
  Background in this file will be sky blue.

  MONITORWIDTH=<nnn>
  This parameter allows you to specify the width in inches of the image on
  your monitor for the purpose of getting the correct stereo effect when
  viewing RDS images. See Random Dot Stereograms (RDS) (p. 82).

                     Fractint Version xx.xx                     Page 120

 5.19 Batch Mode

  It IS possible, believe it or not, to become so jaded with the screen
  drawing process, so familiar with the types and options, that you just
  want to hit a key and do something else until the final images are safe
  on disk.  To do this, start Fractint with the BATCH=yes parameter.  To
  set up a batch run with the parameters required for a particular image
  you might:
    o Find an interesting area.  Note the parameters from the <Tab>
      display.  Then use an editor to write a batch file.
    o Find an interesting area.  Set all the options you'll want in the
      batch run.  Use the <B> command to store the parameters in a file.
      Then use an editor to add the additional required batch mode
      parameters (such as VIDEO=) to the generated parameter file entry.
      Then run the batch using "fractint @myname.par/myentry" (if you told
      the <B> command to use file "myname" and to name the entry
      "myentry").

  Another approach to batch mode calculations, using "FILENAME=" and
  resume, is described later.

  When modifying a parameter file entry generated by the <B> command, the
  only parameters you must add for a batch mode run are "BATCH=yes", and
  "VIDEO=xxx" to select a video mode.  You might want to also add
  "SAVENAME=[name]" to name the result as something other than the default
  FRACT001.GIF.  Or, you might find it easier to leave the generated
  parameter file unchanged and add these parameters by using a command
  like:
     fractint @myname.par/myentry batch=y video=AF3 savename=mygif

  "BATCH=yes" tells Fractint to run in batch mode -- that is, Fractint
  draws the image using whatever other parameters you specified, then acts
  as if you had hit <S> to save the image, then exits to DOS.

  "FILENAME=" can be used with "BATCH=yes" to resume calculation of an
  incomplete image.  For instance, you might interactively find an image
  you like; then select some slow options (a high resolution disk video
  mode, distance estimator method, high maxiter, or whatever);  start the
  calculation;  then interrupt immediately with a <S>ave.  Rename the save
  file (fract001.gif if it is the first in the session and you didn't name
  it with the <X> options or "savename=") to xxx.gif. Later you can run
  Fractint in batch mode to finish the job:
      fractint batch=yes filename=xxx savename=xxx

  "SAVETIME=nnn" is useful with long batch calculations, to store a
  checkpoint every nnn minutes.  If you start a many hour calculation with
  say "savetime=60", and a power failure occurs during the calculation,
  you'll have lost at most an hour of work on the image.  You can resume
  calculation from the save file as above.  Automatic saves triggered by
  SAVETIME do not increment the save file name. The same file is
  overwritten by each auto save until the image completes.  But note that
  Fractint does not directly over-write save files.  Instead, each save
  operation writes a temporary file FRACTINT.TMP, then deletes the prior
  save file, then renames FRACTINT.TMP to be the new save file.  This
  protects against power failures which occur during a save operation - if
  such a power failure occurs, the prior save file is intact and there's a

                     Fractint Version xx.xx                     Page 121

  harmless incomplete FRACTINT.TMP on your disk.

  If you want to spread a many-hour image over multiple bits of free
  machine time you could use a command like:
      fractint batch=yes filename=xxx savename=xxx savetime=60 video=F3
  While this batch is running, hit <S> (almost any key actually) to tell
  fractint to save what it has done so far and give your machine back.  A
  status code of 2 is returned by fractint to the batch file.  Kick off
  the batch again when you have another time slice for it.

  While running a batch file, pressing any key will cause Fractint to exit
  with an errorlevel = 2.  Any error that interrupts an image save to disk
  will cause an exit with errorlevel = 2.  Any error that prevents an
  image from being generated will cause an exit with errorlevel = 1.

  The SAVETIME= parameter, and batch resumes of partial calculations, only
  work with fractal types which can be resumed.  See "Interrupting and
  Resuming" (p. 25) for information about non-resumable types.


 5.20 Browser Parameters

  This Screen enables you to control Fractints built in file browsing
  utility.  If you don't know what that is see Browse Commands (p. 28).
  This screen is selected with <Ctrl-B> from just about anywhere.

  "Autobrowsing"
  Select yes if you want the loaded image to be scanned for sub images
  immediately without pressing 'L' every time.

  "Ask about GIF video mode"
  Allows turning on and off the display of the video mode table when
  loading GIFs.  This has the same effect as the askvideo= command.

  "Type/Parm check"
  Select whether the browser tests for fractal type or parms when deciding
  whether a file is a sub image of the current screen or not. DISABLE WITH
  CAUTION! or things could get confusing. These tests can be switched off
  to allow such situations as wishing to display old images that were
  generated using a formula type which is now implemented as a built in
  fractal type.
  "Confirm deletes"
  Set this to No if you get fed up with the double prompting that the
  browser gives when deleting a file.  It won't get rid of the first
  prompt however.

  "Smallest window"
  This parameter determines how small the image would have to be onscreen
  before it decides not to include it in the selection of files.  The size
  is entered in decimal pixels so, for instance, this could be set to 0.2
  to allow images that are up to around three maximum zooms away
  (depending on the current video resolution) to be loaded instantly.  Set
  this to 0 to enable all sub images to be detected.  This can lead to a
  very cluttered screen!  The primary use is in conjunction with the
  search file mask (see below) to allow location of high magnification
  images within an overall view (like the whole Mset ).

                     Fractint Version xx.xx                     Page 122

  "Smallest box"
  This determines when the image location is shown as crosshairs rather
  than a rather small box.  Set this according to how good your eyesight
  is (probably worse than before you started staring at fractals all the
  time :-)) or the resolution of your screen.  WARNING the crosshairs
  routine centers the cursor on one corner of the image box at the moment
  so this looks misleading if set too large.
  "Search Mask"
  Sets the file name pattern which the browser searches, this can be used
  to search out the location of a file by setting this to the filename and
  setting smallest image to 0 (see above).

                     Fractint Version xx.xx                     Page 123

 6. Hardware Support


 6.1 Notes on Video Modes, "Standard" and Otherwise

  True to the spirit of public-domain programming, Fractint makes only a
  limited attempt to verify that your video adapter can run in the mode
  you specify, or even that an adapter is present, before writing to it.
  So if you use the "video=" command line parameter, check it before using
  a new version of Fractint - the old key combo may now call an
  ultraviolet holographic mode.

  EGA

  Fractint assumes that every EGA adapter has a full 256K of memory (and
  can therefore display 640 x 350 x 16 colors), but does nothing to verify
  that fact before slinging pixels.

  "TWEAKED" VGA MODES

  The IBM VGA adapter is a highly programmable device, and can be set up
  to display many video-mode combinations beyond those "officially"
  supported by the IBM BIOS. E.g. 320x400x256 and 360x480x256 (the latter
  is one of our favorites).  These video modes are perfectly legal, but
  temporarily reprogram the adapter (IBM or fully register-compatible) in
  a non-standard manner that the BIOS does not recognize.

  Fractint also contains code that sets up the IBM (or any truly register-
  compatible) VGA adapter for several extended modes such as 704x528,
  736x552, 768x576, and 800x600. It does this by programming the VGA
  controller to use the fastest dot-clock on the IBM adapter (28.322 MHz),
  throwing more pixels, and reducing the refresh rate to make up for it.

  These modes push many monitors beyond their rated specs, in terms of
  both resolution and refresh rate. Signs that your monitor is having
  problems with a particular "tweaked" mode include:
   o vertical or horizontal overscan (displaying dots beyond the edges of
     your visible CRT area)
   o flickering (caused by a too-slow refresh rate)
   o vertical roll or total garbage on the screen (your monitor simply
     can't keep up, or is attempting to "force" the image into a pre-set
     mode that doesn't fit).

  We have successfully tested the modes up to 768x576 on an IBM PS/2 Model
  80 connected to IBM 8513, IBM 8514, NEC Multisync II, and Zenith 1490
  monitors (all of which exhibit some overscan and flicker at the highest
  rates), and have tested 800x600 mode on the NEC Multisync II (although
  it took some twiddling of the vertical-size control).

  SUPER-EGA AND SUPER-VGA MODES

  Since version 12.0, we've used both John Bridges' SuperVGA Autodetecting
  logic *and* VESA adapter detection, so that many brand-specific SuperVGA
  modes have been combined into single video mode selection entries.
  There is now exactly one entry for SuperVGA 640x480x256 mode, for
  instance.

                     Fractint Version xx.xx                     Page 124

  If Fractint's automatic SuperVGA/VESA detection logic guesses wrong, and
  you know which SuperVGA chipset your video adapter uses, you can use the
  "adapter=" command-line option to force Fractint to assume the presence
  of a specific SuperVGA Chipset - see Video Parameters (p. 111) for
  details.

  8514/A MODES

  The IBM 8514/A modes (640x480 and 1024x768) default to using the
  hardware registers.  If an error occurs when trying to open the adapter,
  an attempt will be made to use IBM's software interface, and requires
  the preloading of IBM's HDILOAD TSR utility.

  The Adex 1280x1024 modes were written for and tested on an Adex
  Corporation 8514/A using a Brooktree DAC.  The ATI GU 800x600x256 and
  1280x1024x16 modes require a ROM bios version of 1.3 or higher for
  800x600 and 1.4 or higher for 1280x1024.

  There are two sets of 8514/A modes: full sets (640x480, 800x600,
  1024x768, 1280x1024) which cover the entire screen and do NOT have a
  border color (so that you cannot tell when you are "paused" in a color-
  cycling mode), and partial sets (632x474, 792x594, 1016x762, 1272x1018)
  with small border areas which do turn white when you are paused in
  color-cycling mode. Also, while these modes are declared to be 256-
  color, if you do not have your 8514/A adapter loaded with its full
  complement of memory you will actually be in 16-color mode. The hardware
  register 16-color modes have not been tested.

  If your 8514/A adapter is not truly register compatible and Fractint
  does not detect this, use of the adapter interface can be forced by
  using afi=y or afi=8514 in your SSTOOLS.INI file.

  Finally, because IBM's adapter interface does not handle drawing single
  pixels very well (we have to draw a 1x1 pixel "box"), generating the
  zoom box when using the interface is excruciatingly slow. Still, it
  works!

  XGA MODES

  The XGA adapter is supported using the VESA/SuperVGA Autodetect modes -
  the XGA looks like just another SuperVGA adapter to Fractint.  The
  supported XGA modes are 640x480x256, 1024x768x16, 1024x768x256,
  800x600x16, and 800x600x256.  Note that the 1024x768x256 mode requires a
  full 1MB of adapter memory, the 1024x768 modes require a high-rez
  monitor, and the 800x600 modes require a multisynching monitor such as
  the NEC 2A.

  TARGA MODES

  TARGA support for Fractint is provided courtesy of Joe McLain and has
  been enhanced with the help of Bruce Goren and Richard Biddle.  To use a
  TARGA board with Fractint, you must define two DOS environment
  variables, "TARGA" and "TARGASET".  The definition of these variables is
  standardized by Truevision; if you have a TARGA board you probably
  already have added "SET" statements for these variables to your
  AUTOEXEC.BAT file.  Be aware that there are a LOT of possible TARGA

                     Fractint Version xx.xx                     Page 125

  configurations, and a LOT of opportunities for a TARGA board and a VGA
  or EGA board to interfere with each other, and we may not have all of
  them smoothed away yet.  Also, the TARGA boards have an entirely
  different color-map scheme than the VGA cards, and at the moment they
  cannot be run through the color-cycling menu. The "MAP=" argument (see
  Color Parameters (p. 106)), however, works with both TARGA and VGA
  boards and enables you to redefine the default color maps with either
  board.

  TARGA+ MODES

  To use the special modes supported for TARGA+ adapters, the TARGAP.SYS
  device driver has to be loaded, and the TPLUS.DAT file (included with
  Fractint) must be in the same directory as Fractint.  The video modes
  with names containing "True Color Autodetect" can be used with the
  Targa+.  You might want to use the command line parameters "tplus=",
  "noninterlaced=", "maxcolorres=", and "pixelzoom=" (see Video Parameters
  (p. 111)) in your SSTOOLS.INI file to modify Fractint's use of the
  adapter.


 6.2 "Disk-Video" Modes

  These "video modes" do not involve a video adapter at all. They use (in
  order or preference) your expanded memory, your extended memory, or your
  disk drive (as file FRACTINT.$$$) to store the fractal image. These
  modes are useful for creating images beyond the capacity of your video
  adapter right up to the current internal limit of 2048 x 2048 x 256,
  e.g. for subsequent printing.  They're also useful for background
  processing under multi-tasking DOS managers - create an image in a disk-
  video mode, save it, then restore it in a real video mode.

  While you are using a disk-video mode, your screen will display text
  information indicating whether memory or your disk drive is being used,
  and what portion of the "screen" is being read from or written to.  A
  "Cache size" figure is also displayed. 64K is the maximum cache size.
  If you see a number less than this, it means that you don't have a lot
  of memory free, and that performance will be less than optimum.  With a
  very low cache size such as 4 or 6k, performance gets considerably worse
  in cases using solid guessing, boundary tracing, plasma, or anything
  else which paints the screen non-linearly.  If you have this problem,
  all we can suggest is having fewer TSR utilities loaded before starting
  Fractint, or changing in your config.sys file, such as reducing a very
  high BUFFERS value.

  The zoom box is disabled during disk-video modes (you couldn't see where
  it is anyway).  So is the orbit display feature.

  Color Cycling (p. 16) can be used during disk-video modes, but only to
  load or save a color palette.

  When using real disk for your disk-video, Fractint previously would not
  generate some "attractor" types (e.g. Lorenz) nor "IFS" images.  These
  stress disk drives with intensive reads and writes, but with the caching
  algorithm performance may be acceptable. Currently Fractint gives you a
  warning message but lets you proceed. You can end the calculation with

                     Fractint Version xx.xx                     Page 126

  <Esc> if you think your hard disk is getting too strenuous a workout.

  When using a real disk, and you are not directing the file to a RAM
  disk, and you aren't using a disk caching program on your machine,
  specifying BUFFERS=10 (or more) in your config.sys file is best for
  performance.  BUFFERS=10,2 or even BUFFERS=10,4 is also good.  It is
  also best to keep your disk relatively "compressed" (or "defragmented")
  if you have a utility to do this.

  In order to use extended memory, you must have HIMEM.SYS or an
  equivalent that supports the XMS 2.0 standard or higher.  Also, you
  can't have a VDISK installed in extended memory.  Himem.sys is
  distributed with Microsoft Windows 286/386 and 3.0.  If you have
  problems using the extended memory, try rebooting with just himem.sys
  loaded and see if that clears up the problem.

  If you are running background disk-video fractals under Windows 3, and
  you don't have a lot of real memory (over 2Mb), you might find it best
  to force Fractint to use real disk for disk-video modes.  (Force this by
  using a .pif file with extended memory and expanded memory set to zero.)
  Try this if your disk goes crazy when generating background images,
  which are supposedly using extended or expanded memory.  This problem
  can occur because, to multi-task, sometimes Windows must page an
  application's expanded or extended memory to disk, in big chunks.
  Fractint's own cached disk access may be faster in such cases.


 6.3 Customized Video Modes, FRACTINT.CFG

  If you have a favorite adapter/video mode that you would like to add to
  Fractint... if you want some new sizes of disk-video modes... if you
  want to remove table entries that do not apply to your system... if you
  want to specify different "textsafe=" options for different video
  modes... relief is here, and without even learning "C"!

  You can do these things by modifying the FRACTINT.CFG file with your
  text editor. Saving a backup copy of FRACTINT.CFG first is of course
  highly recommended!

  Fractint uses a video adapter table for most of what it needs to know
  about any particular adapter/mode combination. The table is loaded from
  FRACTINT.CFG each time Fractint is run. It can contain information for
  up to 300 adapter/mode combinations. The table entries, and the function
  keys they are tied to, are displayed in the "select video mode" screen.

  This table makes adding support for various third-party video cards and
  their modes much easier, at least for the ones that pretend to be
  standard with extra dots and/or colors. There is even a special "roll-
  your-own" video mode (mode 19) enabling those of you with "C" compilers
  and a copy of the Fractint source to generate video modes supporting
  whatever adapter you may have.

  The table as currently distributed begins with nine standard and several
  non-standard IBM video modes that have been exercised successfully with
  a PS/2 model 80. These entries, coupled with the descriptive comments in
  the table definition and the information supplied (or that should have

                     Fractint Version xx.xx                     Page 127

  been supplied!) with your video adapter, should be all you need to add
  your own entries.

  After the IBM and quasi-pseudo-demi-IBM modes, the table contains an
  ever-increasing number of entries for other adapters. Almost all of
  these entries have been added because someone like you sent us spec
  sheets, or modified Fractint to support them and then informed us about
  it.

  Lines in FRACTINT.CFG which begin with a semi-colon are treated as
  comments.  The rest of the lines must have eleven fields separated by
  commas.  The fields are defined as:

  1. Key assignment. F2 to F10, SF1 to SF10, CF1 to CF10, or AF1 to AF10.
     Blank if no key is assigned to the mode.
  2. The name of the adapter/video mode (25 chars max, no leading blanks).
     The adapter is set up for that mode via INT 10H, with:
  3. AX = this,
  4. BX = this,
  5. CX = this, and
  6. DX = this (hey, having all these registers wasn't OUR idea!)
  7. An encoded value describing how to write to your video memory in that
     mode. Currently available codes are:
    1) Use the BIOS (INT 10H, AH=12/13, AL=color) (last resort - SLOW!)
    2) Pretend it's a (perhaps super-res) EGA/VGA
    3) Pretend it's an MCGA
    4) SuperVGA 256-Color mode using the Tseng Labs chipset
    5) SuperVGA 256-Color mode using the Paradise chipset
    6) SuperVGA 256-Color mode using the Video-7 chipset
    7) Non-Standard IBM VGA 360 x 480 x 256-Color mode
    8) SuperVGA 1024x768x16 mode for the Everex chipset
    9) TARGA video modes
   10) HERCULES video mode
   11) Non-Video, i.e. "disk-video"
   12) 8514/A video modes
   13) CGA 320x200x4-color and 640x200x2-color modes
   14) Reserved for Tandy 1000 video modes
   15) SuperVGA 256-Color mode using the Trident chipset
   16) SuperVGA 256-Color mode using the Chips & Tech chipset
   17) SuperVGA 256-Color mode using the ATI VGA Wonder chipset
   18) SuperVGA 256-Color mode using the EVEREX chipset
   19) Roll-your-own video mode (as you've defined it in YOURVID.C)
   20) SuperVGA 1024x768x16 mode for the ATI VGA Wonder chipset
   21) SuperVGA 1024x768x16 mode for the Tseng Labs chipset
   22) SuperVGA 1024x768x16 mode for the Trident chipset
   23) SuperVGA 1024x768x16 mode for the Video 7 chipset
   24) SuperVGA 1024x768x16 mode for the Paradise chipset
   25) SuperVGA 1024x768x16 mode for the Chips & Tech chipset
   26) SuperVGA 1024x768x16 mode for the Everex Chipset
   27) SuperVGA Auto-Detect mode (we poke around looking for your adapter)
   28) VESA modes
   29) True Color Auto-Detect (currently only Targa+ supported)
  Add 100, 200, 300, or 400 to this code to specify an over-ride "textsafe="
   option to be used with the mode.  100=yes, 200=no, 300=bios, 400=save.
   E.g. 428 for a VESA mode with textsafe=save forced.
  8. The number of pixels across the screen (X - 160 to 2048)

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  9. The number of pixels down the screen (Y - 160 to 2048)
  10. The number of available colors (2, 4, 16, or 256)
  11. A comment describing the mode (25 chars max, leading blanks are OK)

  NOTE that the AX, BX, CX, and DX fields use hexadecimal notation
  (fifteen ==> 'f', sixteen ==> '10'), because that's the way most adapter
  documentation describes it. The other fields use standard decimal
  notation.

  If you look closely at the default entries, you will notice that the IBM
  VGA entries labeled "tweaked" and "non standard" have entries in the
  table with AX = BX = CX = 0, and DX = some other number. Those are
  special flags that we used to tell the program to custom-program the VGA
  adapter, and are NOT undocumented BIOS calls. Maybe they should be, but
  they aren't.

  If you have a fancy adapter and a new video mode that works on it, and
  it is not currently supported, PLEASE GET THAT INFORMATION TO US!  We
  will add the video mode to the list on our next release, and give you
  credit for it. Which brings up another point: If you can confirm that a
  particular video adapter/mode works (or that it doesn't), and the
  program says it is UNTESTED, please get that information to us also.
  Thanks in advance!

                     Fractint Version xx.xx                     Page 129

 7. Common Problems

  Of course, Fractint would never stoop to having a "common" problem.
  These notes describe some, ahem, "special situations" which come up
  occasionally and which even we haven't the gall to label as "features".

  Hang during startup:
    There might be a problem with Fractint's video detection logic and
    your particular video adapter. Try running with "fractint adapter=xxx"
    where xxx is cga, ega, egamono, mcga, or vga.  If "adapter=vga" works,
    and you really have a SuperVGA adapter capable of higher video modes,
    there are other "adapter=" options for a number of SuperVGA chipsets -
    please see the full selection in Video Parameters (p. 111) for
    details.  If this solves the problem, create an SSTOOLS.INI file with
    the "adapter=xxx" command in it so that the fix will apply to every
    run.
    Another possible cause:  If you install the latest Fractint in say
    directory "newfrac", then run it from another directory with the
    command "\newfrac\fractint", *and* you have an older version of
    fractint.exe somewhere in your DOS PATH, a silent hang is all you'll
    get.  See the notes under the "Cannot find FRACTINT.EXE message"
    problem for the reason.
    Another possibility: try one of the "textsafe" parameter choices
    described in Video Parameters (p. 111).

  Scrambled image when returning from a text mode display:
    If an image which has been partly or completely generated gets partly
    destroyed when you return to it from the menu, help, or the
    information display, please try the various "textsafe" parameter
    options - see Video Parameters (p. 111) for details.  If this cures
    the problem, create an SSTOOLS.INI file with the "textsafe=xxx"
    command so that the fix will apply to every run.

  "Holes" in an image while it is being drawn:
    Little squares colored in your "inside" color, in a pattern of every
    second square of that size, in solid guessing mode, both across and
    down (i.e., 1 out of 4), are a symptom of an image which should be
    calculated with more conservative periodicity checking than the
    default.  See the Periodicity parameter under Image Calculation
    Parameters (p. 104).

  Black bar at top of screen during color cycling on 8086/8088 machines:
    (This might happen intermittently, not every run.)
    "fractint cyclelimit=10" might cure the problem.  If so, increase the
    cyclelimit value (try increasing by 5 or 10 each time) until the
    problem reappears, then back off one step and add that cyclelimit
    value to your SSTOOLS.INI file.

  Other video problems:

    If you are using a VESA driver with your video adapter, the first
    thing to try is the "vesadetect=no" parameter. If that fixes the
    problem, add it to your SSTOOLS.INI file to make the fix permanent.

                     Fractint Version xx.xx                     Page 130

    It may help to explicitly specify your type of adapter - see the
    "adapter=" parameter in Video Parameters (p. 111).

    We've had one case where a video driver for Windows does not work
    properly with Fractint.  If running under Windows, DesqView, or some
    other layered environment, try running Fractint directly from DOS to
    see if that avoids the problem.
    We've also had one case of a problem co-existing with "386 to the
    Max".

    We've had one report of an EGA adapter which got scrambled images in
    all modes until "textsafe=no" was used (see Video Parameters (p. 111)
    ).

    Also, see Video Adapter Notes (p. 123) for information about enhanced
    video modes - Fractint makes only limited attempts to verify that a
    video mode you request is actually supported by your adapter.

  Other Hangs and Strange Behavior:
    We've had some problems (hangs and solid beeps) on an FPU equipped
    machine when running under Windows 3's enhanced mode.  The only ways
    around the problem we can find are to either run the Fractint image
    involved outside Windows, or to use the DOS command "SET NO87=nofpu"
    before running Fractint.  (This SET command makes Fractint ignore your
    fpu, so things might be a lot slower as a result.)

  Insufficient memory:
    Fractint requires a fair bit of memory to run.  Most machines with at
    least 640k (ok sticklers, make that "PC-compatible machines") will
    have no problem.  Machines with 512k and machines with many TSR
    utilities and/or a LAN interface may have problems.  Some Fractint
    features allocate memory when required during a run.  If you get a
    message about insufficient memory, or suspect that some problem is due
    to a memory shortage, you could try commenting out some TSR utilities
    in your AUTOEXEC.BAT file, some non-critical drivers in your
    CONFIG.SYS file, or reducing the BUFFERS parameter in your CONFIG.SYS.

  "Cannot find FRACTINT.EXE" message:
    Fractint is an overlayed program - some parts of it are brought from
    disk into memory only when used.  The overlay manager needs to know
    where to find the program.  It must be named FRACTINT.EXE (which it is
    unless somebody renamed it), and you should either be in the directory
    containing it when you start Fractint, or that directory should be in
    your DOS PATH.

  "File FRACTINT.CFG is missing or invalid" message:
    You should either start Fractint while you are in the directory
    containing it, or should have that directory in your DOS PATH
    variable.  If that isn't the problem, maybe you have a FRACTINT.CFG
    file from an older release of Fractint lying around? If so, best
    rename or delete it.  If that isn't the problem either, then the
    FRACTINT.CFG included in the FRAINT.EXE release file has probably been
    changed or deleted. Best reinstall Fractint to get a fresh copy.

                     Fractint Version xx.xx                     Page 131

  Some other program doesn't like GIF files created by Fractint:
    Fractint generates nice clean GIF89A spec files, honest! But telling
    this to the other program isn't likely to change its mind. Instead,
    try an option which might get around the problem: run Fractint with
    the command line option "gif87a=yes" and then save an image. Fractint
    will store the image in the older GIF87A format, without any fractal
    parameters in it (so you won't be able to load the image back into
    Fractint and zoom into it - the fractal type, coordinates, etc. are
    not stored in this older format), and without an "aspect ratio" in the
    GIF header (we've seen one utility which doesn't like that field.)

  Disk video mode performance:
    This won't be blindingly fast at the best of times, but there are
    things which can slow it down and can be tuned.  See "Disk-Video"
    Modes (p. 125) for details.

                     Fractint Version xx.xx                     Page 132

 8. Fractals and the PC


 8.1 A Little History


 8.1.1 Before Mandelbrot

  Like new forms of life, new branches of mathematics and science don't
  appear from nowhere. The ideas of fractal geometry can be traced to the
  late nineteenth century, when mathematicians created shapes -- sets of
  points -- that seemed to have no counterpart in nature.  By a wonderful
  irony, the "abstract" mathematics descended from that work has now
  turned out to be MORE appropriate than any other for describing many
  natural shapes and processes.

  Perhaps we shouldn't be surprised.  The Greek geometers worked out the
  mathematics of the conic sections for its formal beauty; it was two
  thousand years before Copernicus and Brahe, Kepler and Newton overcame
  the preconception that all heavenly motions must be circular, and found
  the ellipse, parabola, and hyperbola in the paths of planets, comets,
  and projectiles.

  In the 17th century Newton and Leibniz created calculus, with its
  techniques for "differentiating" or finding the derivative of functions
  -- in geometric terms, finding the tangent of a curve at any given
  point.  True, some functions were discontinuous, with no tangent at a
  gap or an isolated point. Some had singularities: abrupt changes in
  direction at which the idea of a tangent becomes meaningless. But these
  were seen as exceptional, and attention was focused on the "well-
  behaved" functions that worked well in modeling nature.

  Beginning in the early 1870s, though, a 50-year crisis transformed
  mathematical thinking. Weierstrass described a function that was
  continuous but nondifferentiable -- no tangent could be described at any
  point. Cantor showed how a simple, repeated procedure could turn a line
  into a dust of scattered points, and Peano generated a convoluted curve
  that eventually touches every point on a plane. These shapes seemed to
  fall "between" the usual categories of one-dimensional lines, two-
  dimensional planes and three-dimensional volumes. Most still saw them as
  "pathological" cases, but here and there they began to find
  applications.

  In other areas of mathematics, too, strange shapes began to crop up.
  Poincare attempted to analyze the stability of the solar system in the
  1880s and found that the many-body dynamical problem resisted
  traditional methods. Instead, he developed a qualitative approach, a
  "state space" in which each point represented a different planetary
  orbit, and studied what we would now call the topology -- the
  "connectedness" -- of whole families of orbits. This approach revealed
  that while many initial motions quickly settled into the familiar
  curves, there were also strange, "chaotic" orbits that never became
  periodic and predictable.

                     Fractint Version xx.xx                     Page 133

  Other investigators trying to understand fluctuating, "noisy" phenomena
  -- the flooding of the Nile, price series in economics, the jiggling of
  molecules in Brownian motion in fluids -- found that traditional models
  could not match the data. They had to introduce apparently arbitrary
  scaling features, with spikes in the data becoming rarer as they grew
  larger, but never disappearing entirely.

  For many years these developments seemed unrelated, but there were
  tantalizing hints of a common thread. Like the pure mathematicians'
  curves and the chaotic orbital motions, the graphs of irregular time
  series often had the property of self-similarity: a magnified small
  section looked very similar to a large one over a wide range of scales.


 8.1.2 Who Is This Guy, Anyway?

  While many pure and applied mathematicians advanced these trends, it is
  Benoit Mandelbrot above all who saw what they had in common and pulled
  the threads together into the new discipline.

  He was born in Warsaw in 1924, and moved to France in 1935. In a time
  when French mathematical training was strongly analytic, he visualized
  problems whenever possible, so that he could attack them in geometric
  terms.  He attended the Ecole Polytechnique, then Caltech, where he
  encountered the tangled motions of fluid turbulence.

  In 1958 he joined IBM, where he began a mathematical analysis of
  electronic "noise" -- and began to perceive a structure in it, a
  hierarchy of fluctuations of all sizes, that could not be explained by
  existing statistical methods. Through the years that followed, one
  seemingly unrelated problem after another was drawn into the growing
  body of ideas he would come to call fractal geometry.

  As computers gained more graphic capabilities, the skills of his mind's
  eye were reinforced by visualization on display screens and plotters.
  Again and again, fractal models produced results -- series of flood
  heights, or cotton prices -- that experts said looked like "the real
  thing."

  Visualization was extended to the physical world as well. In a
  provocative essay titled "How Long Is the Coast of Britain?" Mandelbrot
  noted that the answer depends on the scale at which one measures: it
  grows longer and longer as one takes into account every bay and inlet,
  every stone, every grain of sand. And he codified the "self-similarity"
  characteristic of many fractal shapes -- the reappearance of
  geometrically similar features at all scales.

  First in isolated papers and lectures, then in two editions of his
  seminal book, he argued that many of science's traditional mathematical
  models are ill-suited to natural forms and processes: in fact, that many
  of the "pathological" shapes mathematicians had discovered generations
  before are useful approximations of tree bark and lung tissue, clouds
  and galaxies.

                     Fractint Version xx.xx                     Page 134

  Mandelbrot was named an IBM Fellow in 1974, and continues to work at the
  IBM Watson Research Center. He has also been a visiting professor and
  guest lecturer at many universities.


 8.2 A Little Code


 8.2.1 Periodicity Logic

  The "Mandelbrot Lake" in the center of the M-set images is the
  traditional bane of plotting programs. It sucks up the most computer
  time because it always reaches the iteration limit -- and yet the most
  interesting areas are invariably right at the edge the lake.  (See The
  Mandelbrot Set (p. 33) for a description of the iteration process.)

  Thanks to Mark Peterson for pointing out (well, he more like beat us
  over the head until we paid attention) that the iteration values in the
  middle of Mandelbrot Lake tend to decay to periodic loops (i.e., Z(n+m)
  == Z(n), a fact that is pointed out on pages 58-61 of "The Beauty of
  Fractals"). An intelligent program (like the one he wrote) would check
  for this periodicity once in a while, recognize that iterations caught
  in a loop are going to max out, and bail out early.

  For speed purposes, the current version of the program turns this
  checking algorithm on only if the last pixel generated was in the lake.
  (The checking itself takes a small amount of time, and the pixels on the
  very edge of the lake tend to decay to periodic loops very slowly, so
  this compromise turned out to be the fastest generic answer).

  Try a full M-set plot with a 1000-iteration maximum with any other
  program, and then try it on this one for a pretty dramatic proof of the
  value of periodicity checking.

  You can get a visual display of the periodicity effects if you press
  <O>rbits while plotting. This toggles display of the intermediate
  iterations during the generation process.  It also gives you an idea of
  how much work your poor little PC is going through for you!  If you use
  this toggle, it's best to disable solid-guessing first using <1> or <2>
  because in its second pass, solid-guessing bypasses many of the pixel
  calculations precisely where the orbits are most interesting.

  Mark was also responsible for pointing out that 16-bit integer math was
  good enough for the first few levels of M/J images, where the round-off
  errors stay well within the area covered by a single pixel. Fractint now
  uses 16-bit math where applicable, which makes a big difference on non-
  32-bit PCs.


 8.2.2 Limitations of Integer Math (And How We Cope)

  By default, Fractint uses 16-bit and/or 32-bit integer math to generate
  nearly all its fractal types. The advantage of integer math is speed:
  this is by far the fastest such plotter that we have ever seen on any
  PC. The disadvantage is an accuracy limit. Integer math represents
  numbers like 1.00 as 32-bit integers of the form [1.00 * (2^29)]

                     Fractint Version xx.xx                     Page 135

  (approximately a range of 500,000,000) for the Mandelbrot and Julia
  sets. Other integer fractal types use a bitshift of 24 rather than 29,
  so 1.0 is stored internally as [1.00 * (2^24)]. This yields accuracy of
  better than 8 significant digits, and works fine... until the initial
  values of the calculations on consecutive pixels differ only in the
  ninth decimal place.

  At that point, if Fractint has a floating-point algorithm handy for that
  particular fractal type (and virtually all of the fractal types have one
  these days), it will silently switch over to the floating-point
  algorithm and keep right on going.  Fair warning - if you don't have an
  FPU, the effect is that of a rocket sled hitting a wall of jello, and
  even if you do, the slowdown is noticeable.

  If it has no floating-point algorithm, Fractint does the best it can: it
  switches to its minimal drawing mode, with adjacent pixels having
  initial values differing by 1 (really 0.000000002).  Attempts to zoom
  further may result in moving the image around a bit, but won't actually
  zoom.  If you are stuck with an integer algorithm, you can reach minimal
  mode with your fifth consecutive "maximum zoom", each of which covers
  about 0.25% of the previous screen. By then your full-screen image is an
  area less than 1/(10^13)th [~0.0000000000001] the area of the initial
  screen.  (If your image is rotated or stretched very slightly, you can
  run into the wall of jello as early as the fourth consecutive maximum
  zoom.  Rotating or stretching by larger amounts has less impact on how
  soon you run into it.)

  Think of it this way: at minimal drawing mode, your VGA display would
  have to have a surface area of over one million square miles just to be
  able to display the entire M-set using the integer algorithms.  Using
  the floating-point algorithms, your display would have to be big enough
  to fit the entire solar system out to the orbit of Saturn inside it.  So
  there's a considerable saving on hardware, electricity and desk space
  involved here.  Also, you don't have to take out asteroid insurance.

  32 bit integers also limit the largest number which can be stored.  This
  doesn't matter much since numbers outside the supported range (which is
  between -4 and +4) produce a boring single color. If you try to zoom-out
  to reduce the entire Mandelbrot set to a speck, or to squeeze it to a
  pancake, you'll find you can't do so in integer math mode.


 8.2.3 Arbitrary Precision and Deep Zooming

  The zoom limit of Fractint is approximately 10^15 (10 to the fifteenth
  power). This limit is due to the precision possible with the computer
  representation of numbers as 64 bit double precision data. To give you
  an idea of just how big a magnification 10^15 is, consider this. At the
  scale of your computer screen while displaying a tiny part of the
  Mandelbrot set at the deepest possible zoom, the entire Mandelbrot set
  would be many millions of miles wide, as big as the orbit of Jupiter.

  Big as this zoom magnification is, your PC can do better using something
  called arbitrary precision math. Instead of using 64 bit double
  precision to represent numbers, your computer software allocates as much
  memory as needed to create a data type supporting as many decimals of

                     Fractint Version xx.xx                     Page 136

  precision as you want.

  Incorporation of this feature in Fractint was inspired by Jay Hill and
  his DEEPZOOM program which uses the shareware MFLOAT programming
  library.  Several of the Stone Soup programmers noticed Jay's posts in
  the Internet sci.fractals newsgroup and began to investigate adding
  arbitrary precision to Fractint. High school math and physics teacher
  Wes Loewer wrote an arbitrary precision library in both 80x86 assembler
  and C, and the Stone Soup team incorporated Wes's library into Fractint.
  Initially, support was added for fractal types mandel, julia, manzpower,
  and julzpower.

  Normally, when you reach Fractint's zoom limit, Fractint simply refuses
  to let you zoom any more. When using the fractal types that support
  arbitrary precision, you will not reach this limit, but can keep on
  zooming. When you pass the threshold between double precision and
  arbitrary precision, Fractint will dramatically slow down. The <tab>
  status screen can be used to verify that Fractint is indeed using
  arbitrary precision.

  Fractals with arbitrary precision are SLOW, as much as ten times slower
  than if the math were done with your math coprocessor, and even slower
  simply because the zoom depth is greater. The good news, if you want to
  call it that, is that your math coprocessor is not needed;
  coprocessorless machines can produce deep zooms with the same glacial
  slowness as machines with coprocessors!

  Maybe the real point of arbitrary precision math is to prolong the
  "olden" days when men were men, women were women, and real fractal
  programmers spent weeks generating fractals. One of your Stone Soup
  authors has a large monitor that blinks a bit when changing video modes-
  -PCs have gotten so fast that Fractint finishes the default 320x200
  Mandelbrot before the monitor can even complete its blinking transition
  to graphics mode! Computers are getting faster every day, and soon a new
  generation of fractal lovers might forget that fractal generation is
  *supposed* to be slow, just as it was in Grandpa's day when they only
  had Pentium chips. The solution to this educational dilemma is
  Fractint's arbitrary precision feature. Even the newest sexium and
  septium machines are going to have to chug for days or weeks at the
  extreme zoom depths now possible ...

  So how far can you zoom? How does 10^1600 sound--roughly 1600 decimal
  digits of precision. To put *this* magnification in perspective, the
  "tiny" ratio of 10^61 is the ratio of the entire visible universe to the
  smallest quantum effects. With 1600 digits to work with, you can expand
  an electron-sized image up to the size of the visible universe, not once
  but more than twenty times. So you can examine screen-sized portions of
  a Mandelbrot set so large all but a tiny part of it would be vastly
  farther away than the billion or so light year limit of our best
  telescopes.

  Lest anyone suppose that we Stone Soupers suffer from an inflated pride
  over having thus spanned the Universe, current inflationary cosmological
  theories estimate the size of the universe to be unimaginably larger
  than the "tiny" part we can see.

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  Note: many of Fractint's options do not work with arbitrary precision.
  To experiment with arbitrary precision at the speedier ordinary
  magnifications, start Fractint with the debug=3200 command-line option.
  With the exception of mandel and manzpower perturbations, values that
  would normally be entered in the Parameters and Coordinates screens need
  to be entered using the command-line interface or .par files.  Other
  known things that do not yet work with arbitrary precision are:
  biomorph, decomp, distance estimator, inversion, Julia-Mandel switch,
  history, orbit-in-window, and the browse feature.


 8.2.4 The Fractint "Fractal Engine" Architecture

  Several of the authors would never ADMIT this, but Fractint has evolved
  a powerful and flexible architecture that makes adding new fractals very
  easy. (They would never admit this because they pride themselves on
  being the sort that mindlessly but happily hacks away at code and "sees
  if it works and doesn't hang the machine".)

  Many fractal calculations work by taking a rectangle in the complex
  plane, and, point by point, calculating a color corresponding to that
  point.  Furthermore, the color calculation is often done by iterating a
  function over and over until some bailout condition is met.  (See The
  Mandelbrot Set (p. 33) for a description of the iteration process.)

  In implementing such a scheme, there are three fractal-specific
  calculations that take place within a framework that is pretty much the
  same for them all.  Rather than copy the same code over and over, we
  created a standard fractal engine that calls three functions that may be
  bolted in temporarily to the engine.  The "bolting in" process uses the
  C language mechanism of variable function pointers.

  These three functions are:

     1) a setup function that is run once per image, to do any required
     initialization of variables,

     2) a once-per-pixel function that does whatever initialization has to
     be done to calculate a color for one pixel, and

     3) a once-per-orbit-iteration function, which is the fundamental
     fractal algorithm that is repeatedly iterated in the fractal
     calculation.

  The common framework that calls these functions can contain all sorts of
  speedups, tricks, and options that the fractal implementor need not
  worry about.  All that is necessary is to write the three functions in
  the correct way, and BINGO! - all options automatically apply. What
  makes it even easier is that usually one can re-use functions 1) and 2)
  written for other fractals, and therefore only need to write function
  3).

  Then it occurred to us that there might be more than one sort of fractal
  engine, so we even allowed THAT to be bolted in. And we created a data
  structure for each fractal that includes pointers to these four
  functions, various prompts, a default region of the complex plane, and

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  various miscellaneous bits of information that allow toggling between
  Julia and Mandelbrot or toggling between the various kinds of math used
  in implementation.

  That sounds pretty flexible, but there is one drawback - you have to be
  a C programmer and have a C compiler to make use of it! So we took it a
  step further, and designed a built-in high level compiler, so that you
  can enter the formulas for the various functions in a formula file in a
  straightforward algebra-like language, and Fractint will compile them
  and bolt them in for you!

  There is a terrible down side to this flexibility.  Fractint users
  everywhere are going berserk. Fractal-inventing creativity is running
  rampant. Proposals for new fractal types are clogging the mail and the
  telephones.

  All we can say is that non-productivity software has never been so
  potent, and we're sorry, it's our fault!

  Fractint was compiled using Microsoft C 7.0 and Microsoft Assembler 6.0,
  using the "Medium" model. Note that the assembler code uses the "C"
  model option added to version 5.1, and must be assembled with the /MX or
  /ML switch to link with the "C" code. Because it has become too large to
  distribute comfortably as a single compressed file, and because many
  downloaders have no intention of ever modifying it, Fractint is now
  distributed as two files: one containing FRACTINT.EXE, auxiliary files
  and this document, and another containing complete source code
  (including a .MAK file and MAKEFRAC.BAT).  See Distribution of Fractint
  (p. 160).

                     Fractint Version xx.xx                     Page 139

 Appendix A Mathematics of the Fractal Types

  SUMMARY OF FRACTAL TYPES

  ant (p. 67)
      Generalized Ant Automaton as described in the July 1994 Scientific
      American. Some ants wander around the screen. A rule string (the first
      parameter) determines the ant's direction. When the type 1 ant leaves a
      cell of color k, it turns right if the kth symbol in the first parameter
      is a 1, or left otherwise. Then the color in the old cell is incremented.
      The 2nd parameter is a maximum iteration to guarantee that the fractal
      will terminate. The 3rd parameter is the number of ants. The 4th is the
      ant type 1 or 2. The 5th parameter determines if the ants wrap the screen
      or stop at the edge.  The 6th parameter is a random seed. You can slow
      down the ants to see them better using the <x> screen Orbit Delay.
  barnsleyj1 (p. 42)
        z(0) = pixel;
        z(n+1) = (z-1)*c if real(z) >= 0, else
        z(n+1) = (z+1)*c
      Two parameters: real and imaginary parts of c
  barnsleyj2 (p. 42)
        z(0) = pixel;
        if real(z(n)) * imag(c) + real(c) * imag(z((n)) >= 0
           z(n+1) = (z(n)-1)*c
        else
           z(n+1) = (z(n)+1)*c
      Two parameters: real and imaginary parts of c
  barnsleyj3 (p. 42)
        z(0) = pixel;
        if real(z(n) > 0 then z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1)
           + i * (2*real(z((n)) * imag(z((n))) else
        z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1 + real(c) * real(z(n))
               + i * (2*real(z((n)) * imag(z((n)) + imag(c) * real(z(n))
      Two parameters: real and imaginary parts of c.
  barnsleym1 (p. 42)
        z(0) = c = pixel;
        if real(z) >= 0 then
          z(n+1) = (z-1)*c
        else
          z(n+1) = (z+1)*c.
      Parameters are perturbations of z(0)
  barnsleym2 (p. 42)
        z(0) = c = pixel;
        if real(z)*imag(c) + real(c)*imag(z) >= 0
          z(n+1) = (z-1)*c
        else
          z(n+1) = (z+1)*c
      Parameters are perturbations of z(0)

  barnsleym3 (p. 42)
        z(0) = c = pixel;
        if real(z(n) > 0 then z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1)
           + i * (2*real(z((n)) * imag(z((n))) else
        z(n+1) = (real(z(n))^2 - imag(z(n))^2 - 1 + real(c) * real(z(n))
           + i * (2*real(z((n)) * imag(z((n)) + imag(c) * real(z(n))
      Parameters are perturbations of z(0)

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  bifurcation (p. 48)
      Pictorial representation of a population growth model.
        Let P = new population, p = oldpopulation, r = growth rate
        The model is: P = p + r*fn(p)*(1-fn(p)).
      Three parameters: Filter Cycles, Seed Population, and Function.

  bif+sinpi (p. 48)
      Bifurcation variation: model is: P = p + r*fn(PI*p).
      Three parameters: Filter Cycles, Seed Population, and Function.

  bif=sinpi (p. 48)
      Bifurcation variation: model is: P = r*fn(PI*p).
      Three parameters: Filter Cycles, Seed Population, and Function.

  biflambda (p. 48)
      Bifurcation variation: model is: P = r*fn(p)*(1-fn(p)).
      Three parameters: Filter Cycles, Seed Population, and Function.

  bifstewart (p. 48)
      Bifurcation variation: model is: P = (r*fn(p)*fn(p)) - 1.
      Three parameters: Filter Cycles, Seed Population, and Function.

  bifmay (p. 48)
      Bifurcation variation: model is: P = r*p / ((1+p)^beta).
      Three parameters: Filter Cycles, Seed Population, and Beta.

  cellular (p. 66)
      One-dimensional cellular automata or line automata.  The type of CA
      is given by kr, where k is the number of different states of the
      automata and r is the radius of the neighborhood.  The next generation
      is determined by the sum of the neighborhood and the specified rule.
      Four parameters: Initial String, Rule, Type, and Starting Row Number.
      For Type = 21, 31, 41, 51, 61, 22, 32, 42, 23, 33, 24, 25, 26, 27
          Rule =  4,  7, 10, 13, 16,  6, 11, 16,  8, 15, 10, 12, 14, 16 digits
  chip (p. 53)
      Chip attractor from Michael Peters - orbit in two dimensions.
        z(0) = y(0) = 0;
        x(n+1) = y(n) - sign(x(n)) * cos(sqr(ln(abs(b*x(n)-c))))
                                   * arctan(sqr(ln(abs(c*x(n)-b))))
        y(n+1) = a - x(n)
      Parameters are a, b, and c.
  circle (p. 40)
      Circle pattern by John Connett
        x + iy = pixel
        z = a*(x^2 + y^2)
        c = integer part of z
        color = c modulo(number of colors)
  cmplxmarksjul (p. 46)
      A generalization of the marksjulia fractal.
        z(0) = pixel;
        z(n+1) = (c^exp-1)*z(n)^2 + c.
      Four parameters: real and imaginary parts of c,
      and real and imaginary parts of exponent.
  cmplxmarksmand (p. 46)
      A generalization of the marksmandel fractal.
        z(0) = c = pixel;

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        z(n+1) = (c^exp-1)*z(n)^2 + c.
      Four parameters: real and imaginary parts of perturbation
      of z(0), and real and imaginary parts of exponent.

  complexnewton, complexbasin (p. 38)
      Newton fractal types extended to complex degrees. Complexnewton
      colors pixels according to the number of iterations required to
      escape to a root. Complexbasin colors pixels according to which
      root captures the orbit. The equation is based on the newton
      formula for solving the equation z^p = r
        z(0) = pixel;
        z(n+1) = ((p - 1) * z(n)^p + r)/(p * z(n)^(p - 1)).
      Four parameters: real & imaginary parts of degree p and root r.

  diffusion (p. 58)
      Diffusion Limited Aggregation.  Randomly moving points
      accumulate.  Two parameters: border width (default 10), type.

  dynamic (p. 64)
      Time-discrete dynamic system.
        x(0) = y(0) = start position.
        y(n+1) = y(n) + f( x(n) )
        x(n+1) = x(n) - f( y(n) )
        f(k) = sin(k + a*fn1(b*k))
      For implicit Euler approximation: x(n+1) = x(n) - f( y(n+1) )
      Five parameters: start position step, dt, a, b, and the function fn1.

  fn(z)+fn(pix) (p. 47)
        c = z(0) = pixel;
        z(n+1) = fn1(z) + p*fn2(c)
      Six parameters: real and imaginary parts of the perturbation
      of z(0) and factor p, and the functions fn1, and fn2.
  fn(z*z) (p. 47)
        z(0) = pixel;
        z(n+1) = fn(z(n)*z(n))
      One parameter: the function fn.

  fn*fn (p. 47)
        z(0) = pixel; z(n+1) = fn1(n)*fn2(n)
      Two parameters: the functions fn1 and fn2.

  fn*z+z (p. 47)
        z(0) = pixel; z(n+1) = p1*fn(z(n))*z(n) + p2*z(n)
      Five parameters: the real and imaginary components of
      p1 and p2, and the function fn.

  fn+fn (p. 47)
        z(0) = pixel;
        z(n+1) = p1*fn1(z(n))+p2*fn2(z(n))
      Six parameters: The real and imaginary components of
      p1 and p2, and the functions fn1 and fn2.

  formula (p. 55)
      Formula interpreter - write your own formulas as text files!

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  frothybasin (p. 69)
      Pixel color is determined by which attractor captures the orbit.  The
      shade of color is determined by the number of iterations required to
      capture the orbit.
        Z(0) = pixel;  Z(n+1) = Z(n)^2 - C*conj(Z(n))
        where C = 1 + A*i, critical value of A = 1.028713768218725...

  gingerbread (p. 53)
      Orbit in two dimensions defined by:
        x(n+1) = 1 - y(n) + |x(n)|
        y(n+1) = x(n)
      Two parameters: initial values of x(0) and y(0).

  halley (p. 63)
        Halley map for the function: F = z(z^a - 1) = 0
        z(0) = pixel;
        z(n+1) = z(n) - R * F / [F' - (F" * F / 2 * F')]
        bailout when: abs(mod(z(n+1)) - mod(z(n)) < epsilon
      Four parameters: order a, real part of R, epsilon,
         and imaginary part of R.
  henon (p. 52)
      Orbit in two dimensions defined by:
        x(n+1) = 1 + y(n) - a*x(n)*x(n)
        y(n+1) = b*x(n)
      Two parameters: a and b

  hopalong (p. 53)
      Hopalong attractor by Barry Martin - orbit in two dimensions.
        z(0) = y(0) = 0;
        x(n+1) = y(n) - sign(x(n))*sqrt(abs(b*x(n)-c))
        y(n+1) = a - x(n)
      Parameters are a, b, and c.

  hypercomplex (p. 66)
      HyperComplex Mandelbrot set.
        h(0)   = (0,0,0,0)
        h(n+1) = fn(h(n)) + C.
        where "fn" is sin, cos, log, sqr etc.
      Two parameters: cj, ck
      C = (xpixel,ypixel,cj,ck)

  hypercomplexj (p. 66)
      HyperComplex Julia set.
        h(0)   = (xpixel,ypixel,zj,zk)
        h(n+1) = fn(h(n)) + C.
        where "fn" is sin, cos, log, sqr etc.
      Six parameters: c1, ci, cj, ck
      C = (c1,ci,cj,ck)

  icon, icon3d (p. 54)
      Orbit in three dimensions defined by:
        p = lambda + alpha * magnitude + beta * (x(n)*zreal - y(n)*zimag)
        x(n+1) = p * x(n) + gamma * zreal - omega * y(n)
        y(n+1) = p * y(n) - gamma * zimag + omega * x(n)
        (3D version uses magnitude for z)
        Parameters:  Lambda, Alpha, Beta, Gamma, Omega, and Degree

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  IFS (p. 43)
      Barnsley IFS (Iterated Function System) fractals. Apply
      contractive affine mappings.

  julfn+exp (p. 45)
      A generalized Clifford Pickover fractal.
        z(0) = pixel;
        z(n+1) = fn(z(n)) + e^z(n) + c.
      Three parameters: real & imaginary parts of c, and fn

  julfn+zsqrd (p. 45)
        z(0) = pixel;
        z(n+1) = fn(z(n)) + z(n)^2 + c
      Three parameters: real & imaginary parts of c, and fn

  julia (p. 34)
      Classic Julia set fractal.
        z(0) = pixel; z(n+1) = z(n)^2 + c.
      Two parameters: real and imaginary parts of c.

  julia_inverse (p. 36)
      Inverse Julia function - "orbit" traces Julia set in two dimensions.
        z(0) = a point on the Julia Set boundary; z(n+1) = +- sqrt(z(n) - c)
      Parameters: Real and Imaginary parts of c
             Maximum Hits per Pixel (similar to max iters)
             Breadth First, Depth First or Random Walk Tree Traversal
             Left or Right First Branching (in Depth First mode only)
          Try each traversal method, keeping everything else the same.
          Notice the differences in the way the image evolves.  Start with
          a fairly low Maximum Hit limit, then increase it.  The hit limit
          cannot be higher than the maximum colors in your video mode.

  julia(fn||fn) (p. 63)
        z(0) = pixel;
        if modulus(z(n)) < shift value, then
           z(n+1) = fn1(z(n)) + c,
        else
           z(n+1) = fn2(z(n)) + c.
      Five parameters: real, imaginary portions of c, shift value,
                       fn1 and fn2.

  julia4 (p. 45)
      Fourth-power Julia set fractals, a special case
      of julzpower kept for speed.
        z(0) = pixel;
        z(n+1) = z(n)^4 + c.
      Two parameters: real and imaginary parts of c.

  julibrot (p. 57)
      'Julibrot' 4-dimensional fractals.

  julzpower (p. 45)
        z(0) = pixel;
        z(n+1) = z(n)^m + c.
      Three parameters: real & imaginary parts of c, exponent m

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  julzzpwr (p. 45)
        z(0) = pixel;
        z(n+1) = z(n)^z(n) + z(n)^m + c.
      Three parameters: real & imaginary parts of c, exponent m

  kamtorus, kamtorus3d (p. 48)
      Series of orbits superimposed.
      3d version has 'orbit' the z dimension.
        x(0) = y(0) = orbit/3;
        x(n+1) = x(n)*cos(a) + (x(n)*x(n)-y(n))*sin(a)
        y(n+1) = x(n)*sin(a) - (x(n)*x(n)-y(n))*cos(a)
      After each orbit, 'orbit' is incremented by a step size.
      Parameters: a, step size, stop value for 'orbit', and
      points per orbit.

  lambda (p. 39)
      Classic Lambda fractal. 'Julia' variant of Mandellambda.
        z(0) = pixel;
        z(n+1) = lambda*z(n)*(1 - z(n)).
      Two parameters: real and imaginary parts of lambda.

  lambdafn (p. 41)
        z(0) = pixel;
        z(n+1) = lambda * fn(z(n)).
      Three parameters: real, imag portions of lambda, and fn

  lambda(fn||fn) (p. 63)
        z(0) = pixel;
        if modulus(z(n)) < shift value, then
           z(n+1) = lambda * fn1(z(n)),
        else
           z(n+1) = lambda * fn2(z(n)).
      Five parameters: real, imaginary portions of lambda, shift value,
                       fn1 and fn2.

  lorenz, lorenz3d (p. 51)
      Lorenz two lobe attractor - orbit in three dimensions.
      In 2d the x and y components are projected to form the image.
        z(0) = y(0) = z(0) = 1;
        x(n+1) = x(n) + (-a*x(n)*dt) + (   a*y(n)*dt)
        y(n+1) = y(n) + ( b*x(n)*dt) - (     y(n)*dt) - (z(n)*x(n)*dt)
        z(n+1) = z(n) + (-c*z(n)*dt) + (x(n)*y(n)*dt)
      Parameters are dt, a, b, and c.
  lorenz3d1 (p. 51)
      Lorenz one lobe attractor - orbit in three dimensions.
      The original formulas were developed by Rick Miranda and Emily Stone.
        z(0) = y(0) = z(0) = 1; norm = sqrt(x(n)^2 + y(n)^2)
        x(n+1) = x(n) + (-a*dt-dt)*x(n) + (a*dt-b*dt)*y(n)
           + (dt-a*dt)*norm + y(n)*dt*z(n)
        y(n+1) = y(n) + (b*dt-a*dt)*x(n) - (a*dt+dt)*y(n)
           + (b*dt+a*dt)*norm - x(n)*dt*z(n) - norm*z(n)*dt
        z(n+1) = z(n) +(y(n)*dt/2) - c*dt*z(n)
      Parameters are dt, a, b, and c.
  lorenz3d3 (p. 51)
      Lorenz three lobe attractor - orbit in three dimensions.
      The original formulas were developed by Rick Miranda and Emily Stone.

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        z(0) = y(0) = z(0) = 1; norm = sqrt(x(n)^2 + y(n)^2)
        x(n+1) = x(n) +(-(a*dt+dt)*x(n) + (a*dt-b*dt+z(n)*dt)*y(n))/3
            + ((dt-a*dt)*(x(n)^2-y(n)^2)
            + 2*(b*dt+a*dt-z(n)*dt)*x(n)*y(n))/(3*norm)
        y(n+1) = y(n) +((b*dt-a*dt-z(n)*dt)*x(n) - (a*dt+dt)*y(n))/3
            + (2*(a*dt-dt)*x(n)*y(n)
            + (b*dt+a*dt-z(n)*dt)*(x(n)^2-y(n)^2))/(3*norm)
        z(n+1) = z(n) +(3*x(n)*dt*x(n)*y(n)-y(n)*dt*y(n)^2)/2 - c*dt*z(n)
      Parameters are dt, a, b, and c.
  lorenz3d4 (p. 51)
      Lorenz four lobe attractor - orbit in three dimensions.
      The original formulas were developed by Rick Miranda and Emily Stone.
        z(0) = y(0) = z(0) = 1;
        x(n+1) = x(n) +(-a*dt*x(n)^3
           + (2*a*dt+b*dt-z(n)*dt)*x(n)^2*y(n) + (a*dt-2*dt)*x(n)*y(n)^2
           + (z(n)*dt-b*dt)*y(n)^3) / (2 * (x(n)^2+y(n)^2))
        y(n+1) = y(n) +((b*dt-z(n)*dt)*x(n)^3 + (a*dt-2*dt)*x(n)^2*y(n)
           + (-2*a*dt-b*dt+z(n)*dt)*x(n)*y(n)^2
           - a*dt*y(n)^3) / (2 * (x(n)^2+y(n)^2))
        z(n+1) = z(n) +(2*x(n)*dt*x(n)^2*y(n) - 2*x(n)*dt*y(n)^3 - c*dt*z(n))
      Parameters are dt, a, b, and c.

  lsystem (p. 60)
      Using a turtle-graphics control language and starting with
      an initial axiom string, carries out string substitutions the
      specified number of times (the order), and plots the resulting.

  lyapunov (p. 62)
      Derived from the Bifurcation fractal, the Lyapunov plots the Lyapunov
      Exponent for a population model where the Growth parameter varies between
      two values in a periodic manner.

  magnet1j (p. 59)
        z(0) = pixel;
                  [  z(n)^2 + (c-1)  ] 2
        z(n+1) =  | ---------------- |
                  [  2*z(n) + (c-2)  ]
      Parameters: the real and imaginary parts of c

  magnet1m (p. 59)
        z(0) = 0; c = pixel;
                  [  z(n)^2 + (c-1)  ] 2
        z(n+1) =  | ---------------- |
                  [  2*z(n) + (c-2)  ]
      Parameters: the real & imaginary parts of perturbation of z(0)

  magnet2j (p. 59)
        z(0) = pixel;
                  [  z(n)^3 + 3*(C-1)*z(n) + (C-1)*(C-2)         ] 2
        z(n+1) =  |  -------------------------------------------- |
                  [  3*(z(n)^2) + 3*(C-2)*z(n) + (C-1)*(C-2) + 1 ]
      Parameters: the real and imaginary parts of c
  magnet2m (p. 59)
        z(0) = 0; c = pixel;
                  [  z(n)^3 + 3*(C-1)*z(n) + (C-1)*(C-2)         ] 2
        z(n+1) =  |  -------------------------------------------- |

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                  [  3*(z(n)^2) + 3*(C-2)*z(n) + (C-1)*(C-2) + 1 ]
      Parameters: the real and imaginary parts of perturbation of z(0)

  mandel (p. 33)
      Classic Mandelbrot set fractal.
        z(0) = c = pixel;
        z(n+1) = z(n)^2 + c.
      Two parameters: real & imaginary perturbations of z(0)

  mandel(fn||fn) (p. 63)
        c = pixel;
        z(0) = p1
        if modulus(z(n)) < shift value, then
           z(n+1) = fn1(z(n)) + c,
        else
           z(n+1) = fn2(z(n)) + c.
      Five parameters: real, imaginary portions of p1, shift value,
                       fn1 and fn2.

  mandelcloud (p. 65)
      Displays orbits of Mandelbrot set:
        z(0) = c = pixel;
        z(n+1) = z(n)^2 + c.
      One parameter: number of intervals

  mandel4 (p. 45)
      Special case of mandelzpower kept for speed.
        z(0) = c = pixel;
        z(n+1) = z(n)^4 + c.
      Parameters: real & imaginary perturbations of z(0)

  mandelfn (p. 42)
        z(0) = c = pixel;
        z(n+1) = c*fn(z(n)).
      Parameters: real & imaginary perturbations of z(0), and fn

  manlam(fn||fn) (p. 63)
        c = pixel;
        z(0) = p1
        if modulus(z(n)) < shift value, then
           z(n+1) = fn1(z(n)) * c, else
           z(n+1) = fn2(z(n)) * c.
      Five parameters: real, imaginary parts of p1, shift value, fn1, fn2.

  Martin (p. 53)
      Attractor fractal by Barry Martin - orbit in two dimensions.
        z(0) = y(0) = 0;
        x(n+1) = y(n) - sin(x(n))
        y(n+1) = a - x(n)
      Parameter is a (try a value near pi)

  mandellambda (p. 39)
        z(0) = .5; lambda = pixel;
        z(n+1) = lambda*z(n)*(1 - z(n)).
      Parameters: real & imaginary perturbations of z(0)
  mandphoenix (p. 68)

                     Fractint Version xx.xx                     Page 147

      z(0) = c = pixel, y(0) = 0;
      For degree = 0:
        z(n+1) = z(n)^2 + c.x + c.y*y(n), y(n+1) = z(n)
      For degree >= 2:
        z(n+1) = z(n)^degree + c.x*z(n)^(degree-1) + c.y*y(n)
        y(n+1) = z(n)
      For degree <= -3:
        z(n+1) = z(n)^|degree| + c.x*z(n)^(|degree|-2) + c.y*y(n)
        y(n+1) = z(n)
      Three parameters: real & imaginary perturbations of z(0), and degree.

  mandphoenixclx (p. 68)
      z(0) = c = pixel, y(0) = 0;
      For degree = 0:
        z(n+1) = z(n)^2 + c + p2*y(n), y(n+1) = z(n)
      For degree >= 2:
        z(n+1) = z(n)^degree + c*z(n)^(degree-1) + p2*y(n), y(n+1) = z(n)
      For degree <= -3:
        z(n+1) = z(n)^|degree| + c*z(n)^(|degree|-2) + p2*y(n), y(n+1) = z(n)
      Five parameters: real & imaginary perturbations of z(0), real &
        imaginary parts of p2, and degree.
  manfn+exp (p. 45)
      'Mandelbrot-Equivalent' for the julfn+exp fractal.
        z(0) = c = pixel;
        z(n+1) = fn(z(n)) + e^z(n) + C.
      Parameters: real & imaginary perturbations of z(0), and fn

  manfn+zsqrd (p. 45)
      'Mandelbrot-Equivalent' for the Julfn+zsqrd fractal.
        z(0) = c = pixel;
        z(n+1) = fn(z(n)) + z(n)^2 + c.
      Parameters: real & imaginary perturbations of z(0), and fn

  manowar (p. 47)
        c = z1(0) = z(0) = pixel;
        z(n+1) = z(n)^2 + z1(n) + c;
        z1(n+1) = z(n);
      Parameters: real & imaginary perturbations of z(0)
  manowarj (p. 47)
        z1(0) = z(0) = pixel;
        z(n+1) = z(n)^2 + z1(n) + c;
        z1(n+1) = z(n);
      Parameters: real & imaginary parts of c

  manzpower (p. 45)
      'Mandelbrot-Equivalent' for julzpower.
        z(0) = c = pixel;
        z(n+1) = z(n)^exp + c; try exp = e = 2.71828...
      Parameters: real & imaginary perturbations of z(0), real &
      imaginary parts of exponent exp.

  manzzpwr (p. 45)
      'Mandelbrot-Equivalent' for the julzzpwr fractal.
        z(0) = c = pixel
        z(n+1) = z(n)^z(n) + z(n)^exp + C.
      Parameters: real & imaginary perturbations of z(0), and exponent

                     Fractint Version xx.xx                     Page 148

  marksjulia (p. 46)
      A variant of the julia-lambda fractal.
        z(0) = pixel;
        z(n+1) = (c^exp-1)*z(n)^2 + c.
      Parameters: real & imaginary parts of c, and exponent

  marksmandel (p. 46)
      A variant of the mandel-lambda fractal.
        z(0) = c = pixel;
        z(n+1) = (c^exp-1)*z(n)^2 + c.
      Parameters: real & imaginary parts of perturbations of z(0),
      and exponent

  marksmandelpwr (p. 46)
      The marksmandelpwr formula type generalized (it previously
      had fn=sqr hard coded).
        z(0) = pixel, c = z(0) ^ (z(0) - 1):
        z(n+1) = c * fn(z(n)) + pixel,
      Parameters: real and imaginary perturbations of z(0), and fn

  newtbasin (p. 37)
      Based on the Newton formula for finding the roots of z^p - 1.
      Pixels are colored according to which root captures the orbit.
        z(0) = pixel;
        z(n+1) = ((p-1)*z(n)^p + 1)/(p*z(n)^(p - 1)).
      Two parameters: the polynomial degree p, and a flag to turn
      on color stripes to show alternate iterations.

  newton (p. 38)
      Based on the Newton formula for finding the roots of z^p - 1.
      Pixels are colored according to the iteration when the orbit
      is captured by a root.
        z(0) = pixel;
        z(n+1) = ((p-1)*z(n)^p + 1)/(p*z(n)^(p - 1)).
      One parameter: the polynomial degree p.

  phoenix (p. 68)
      z(0) = pixel, y(0) = 0;
      For degree = 0: z(n+1) = z(n)^2 + p1.x + p2.x*y(n), y(n+1) = z(n)
      For degree >= 2:
       z(n+1) = z(n)^degree + p1.x*z(n)^(degree-1) + p2.x*y(n), y(n+1) = z(n)
      For degree <= -3:
       z(n+1) = z(n)^|degree| + p1.x*z(n)^(|degree|-2) + p2.x*y(n), y(n+1) = z(n)
      Three parameters: real parts of p1 & p2, and degree.

  phoenixcplx (p. 68)
      z(0) = pixel, y(0) = 0;
      For degree = 0: z(n+1) = z(n)^2 + p1 + p2*y(n), y(n+1) = z(n)
      For degree >= 2:
        z(n+1) = z(n)^degree + p1*z(n)^(degree-1) + p2*y(n), y(n+1) = z(n)
      For degree <= -3:
        z(n+1) = z(n)^|degree| + p1*z(n)^(|degree|-2) + p2*y(n), y(n+1) = z(n)
      Five parameters: real & imaginary parts of p1 & p2, and degree.
  pickover (p. 53)
      Orbit in three dimensions defined by:
        x(n+1) = sin(a*y(n)) - z(n)*cos(b*x(n))

                     Fractint Version xx.xx                     Page 149

        y(n+1) = z(n)*sin(c*x(n)) - cos(d*y(n))
        z(n+1) = sin(x(n))
      Parameters: a, b, c, and d.
  plasma (p. 40)
      Random, cloud-like formations.  Requires 4 or more colors.
      A recursive algorithm repeatedly subdivides the screen and
      colors pixels according to an average of surrounding pixels
      and a random color, less random as the grid size decreases.
      Four parameters: 'graininess' (.5 to 50, default = 2), old/new
      algorithm, seed value used, 16-bit out output selection.

  popcorn (p. 46)
      The orbits in two dimensions defined by:
        x(0) = xpixel, y(0) = ypixel;
        x(n+1) = x(n) - h*sin(y(n) + tan(3*y(n))
        y(n+1) = y(n) - h*sin(x(n) + tan(3*x(n))
      are plotted for each screen pixel and superimposed.
      One parameter: step size h.

  popcornjul (p. 46)
      Conventional Julia using the popcorn formula:
        x(0) = xpixel, y(0) = ypixel;
        x(n+1) = x(n) - h*sin(y(n) + tan(3*y(n))
        y(n+1) = y(n) - h*sin(x(n) + tan(3*x(n))
      One parameter: step size h.

  quadruptwo (p. 53)
      Quadruptwo attractor from Michael Peters - orbit in two dimensions.
        z(0) = y(0) = 0;
        x(n+1) = y(n) - sign(x(n)) * sin(ln(abs(b*x(n)-c)))
                                   * arctan(sqr(ln(abs(c*x(n)-b))))
        y(n+1) = a - x(n)
      Parameters are a, b, and c.
  quatjul (p. 65)
      Quaternion Julia set.
        q(0)   = (xpixel,ypixel,zj,zk)
        q(n+1) = q(n)*q(n) + c.
      Four parameters: c, ci, cj, ck
      c = (c1,ci,cj,ck)

  quat (p. 65)
      Quaternion Mandelbrot set.
        q(0)   = (0,0,0,0)
        q(n+1) = q(n)*q(n) + c.
      Two parameters: cj,ck
      c = (xpixel,ypixel,cj,ck)

  rossler3D (p. 52)
      Orbit in three dimensions defined by:
        x(0) = y(0) = z(0) = 1;
        x(n+1) = x(n) - y(n)*dt -   z(n)*dt
        y(n+1) = y(n) + x(n)*dt + a*y(n)*dt
        z(n+1) = z(n) + b*dt + x(n)*z(n)*dt - c*z(n)*dt
      Parameters are dt, a, b, and c.
  sierpinski (p. 44)
      Sierpinski gasket - Julia set producing a 'Swiss cheese triangle'

                     Fractint Version xx.xx                     Page 150

        z(n+1) = (2*x,2*y-1) if y > .5;
            else (2*x-1,2*y) if x > .5;
            else (2*x,2*y)
      No parameters.

  spider (p. 47)
        c(0) = z(0) = pixel;
        z(n+1) = z(n)^2 + c(n);
        c(n+1) = c(n)/2 + z(n+1)
      Parameters: real & imaginary perturbation of z(0)

  sqr(1/fn) (p. 47)
        z(0) = pixel;
        z(n+1) = (1/fn(z(n))^2
      One parameter: the function fn.

  sqr(fn) (p. 47)
        z(0) = pixel;
        z(n+1) = fn(z(n))^2
      One parameter: the function fn.
  test (p. 54)
      'test' point letting us (and you!) easily add fractal types via
      the c module testpt.c.  Default set up is a mandelbrot fractal.
      Four parameters: user hooks (not used by default testpt.c).

  tetrate (p. 47)
        z(0) = c = pixel;
        z(n+1) = c^z(n)
      Parameters: real & imaginary perturbation of z(0)

  threeply (p. 53)
      Threeply attractor by Michael Peters - orbit in two dimensions.
        z(0) = y(0) = 0;
        x(n+1) = y(n) - sign(x(n)) * (abs(sin(x(n))*cos(b)
                                      +c-x(n)*sin(a+b+c)))
        y(n+1) = a - x(n)
      Parameters are a, b, and c.
  tim's_error (p. 46)
      A serendipitous coding error in marksmandelpwr brings to life
      an ancient pterodactyl!  (Try setting fn to sqr.)
        z(0) = pixel, c = z(0) ^ (z(0) - 1):
        tmp = fn(z(n))
        real(tmp) = real(tmp) * real(c) - imag(tmp) * imag(c);
        imag(tmp) = real(tmp) * imag(c) - imag(tmp) * real(c);
        z(n+1) = tmp + pixel;
      Parameters: real & imaginary perturbations of z(0) and function fn

  unity (p. 47)
        z(0) = pixel;
        x = real(z(n)), y = imag(z(n))
        One = x^2 + y^2;
        y = (2 - One) * x;
        x = (2 - One) * y;
        z(n+1) = x + i*y
      No parameters.

                     Fractint Version xx.xx                     Page 151

  INSIDE=BOF60|BOF61|ZMAG|PERIOD

  Here is an *ATTEMPTED* explanation of what the inside=bof60 and
  inside=bof61 options do. This explanation is hereby dedicated to Adrian
  Mariano, who badgered it out of us! For the *REAL* explanation, see
  "Beauty of Fractals", page 62.

  Let p(z) be the function that is repeatedly iterated to generate a
  fractal using the escape-time algorithm.  For example, p(z) = z^2+c in
  the case of a Julia set. Then let pk(z) be the result of iterating the
  function p for k iterations. (The "k" should be shown as a superscript.)
  We could also use the notation pkc(z) when the function p has a
  parameter c, as it does in our example.  Now hold your breath and get
  your thinking cap on. Define a(c) = inf{|pkc(0)|:k=1,2,3,...}. In
  English - a(c) is the greatest lower bound of the images of zero of as
  many iterations as you like. Put another way, a(c) is the closest to the
  origin any point in the orbit starting with 0 gets. Then the index (c)
  is the value of k (the iteration) when that closest point was achieved.
  Since there may be more than one, index(c) is the least such. Got it?
  Good, because the "Beauty of Fractals" explanation of this, is, ahhhh,
  *TERSE* ! Now for the punch line. Inside=bof60 colors the lake
  alternating shades according to the level sets of a(c).  Each band
  represents solid areas of the fractal where the closest value of the
  orbit to the origin is the same.  Inside=bof61 show domains where
  index(c) is constant.  That is, areas where the iteration when the orbit
  swooped closest to the origin has the same value.  Well, folks, that's
  the best we can do! Improved explanations will be accepted for the next
  edition!

  In response to this request for lucidity, Herb Savage offers this
  explanation the bof60 and bof61 options:

   The picture on page 60 of The Beauty of Fractals shows the distance to
   origin of the closest point to the origin in the sequence of points
   generated from a given X,Y coordinate.  The picture on page 61 shows
   the index (or number) in the sequence of the closest point.

  inside=zmag is similar. This option colors inside pixels according to
  the magnitude of the orbit point when maxiter was reached, using the
  formula color = (x^2 + y^2) * maxiter/2 + 1.

  inside=period colors pixels according to the length of their eventual
  cycle.  For example, points that approach a fixed point have color=1.
  Points that approach a 2-cycle have color=2.  Points that do not
  approach a cycle during the iterations performed have color=maxit.  This
  option works best with a fairly large number of iterations.

  INSIDE=EPSCROSS|STARTRAIL

  Kenneth Hooper has written a paper entitled "A Note On Some Internal
  Structures Of The Mandelbrot Set" published in "Computers and Graphics",
  Vol 15, No.2, pp. 295-297.  In that article he describes Clifford
  Pickover's "epsilon cross" method which creates some mysterious plant-
  like tendrils in the Mandelbrot set. The algorithm is this. In the
  escape-time calculation of a fractal, if the orbit comes within .01 of
  the Y-axis, the orbit is terminated and the pixel is colored green.

                     Fractint Version xx.xx                     Page 152

  Similarly, the pixel is colored yellow if it approaches the X-axis.
  Strictly speaking, this is not an "inside" option because a point
  destined to escape could be caught by this bailout criterion.

  Hooper has another coloring scheme called "star trails" that involves
  detecting clusters of points being traversed by the orbit. A table of
  tangents of each orbit point is built, and the pixel colored according
  to how many orbit points are near the first one before the orbit flies
  out of the cluster.  This option looks fine with maxiter=16, which
  greatly speeds the calculation.

  Both of these options should be tried with the outside color fixed
  (outside=<nnn>) so that the "lake" structure revealed by the algorithms
  can be more clearly seen. Epsilon Cross is fun to watch with boundary
  tracing turned on - even though the result is incorrect it is
  interesting! Shucks - what does "incorrect" mean in chaos theory
  anyway?!

  FINITE ATTRACTORS

  Many of Fractint's fractals involve the iteration of functions of
  complex numbers until some "bailout" value is exceeded, then coloring
  the associated pixel according to the number of iterations performed.
  This process identifies which values tend to infinity when iterated, and
  gives us a rough measure of how "quickly" they get there.

  In dynamical terms, we say that "Infinity is an Attractor", as many
  initial values get "attracted" to it when iterated.  The set of all
  points that are attracted to infinity is termed The Basin of Attraction
  of Infinity.  The coloring algorithm used divides this Basin of
  Attraction into many distinct sets, each a single band of one color,
  representing all the points that are "attracted" to Infinity at the same
  "rate".  These sets (bands of color) are termed "Level Sets" - all
  points in such a set are at the same "Level" away from the attractor, in
  terms of numbers of iterations required to exceed the bailout value.

  Thus, Fractint produces colored images of the Level Sets of the Basin of
  Attraction of Infinity, for all fractals that iterate functions of
  Complex numbers, at least.  Now we have a sound mathematical definition
  of what Fractint's "bailout" processing generates, and we have formally
  introduced the terms Attractor, Basin of Attraction, and Level Set, so
  you should have little trouble following the rest of this section!

  For certain Julia-type fractals, Fractint can also display the Level
  Sets of Basins of Attraction of Finite Attractors.  This capability is a
  by-product of the implementation of the MAGNETic fractal types, which
  always have at least one Finite Attractor.

  This option can be invoked by setting the "Look for finite attractor"
  option on the <Y> options screen, or by giving the "finattract=yes"
  command-line option.

  Most Julia-types that have a "lake" (normally colored blue by default)
  have a Finite Attractor within this lake, and the lake turns out to be,
  quite appropriately, the Basin of Attraction of this Attractor.

                     Fractint Version xx.xx                     Page 153

  The "finattract=yes" option (command-line or <Y> options screen)
  instructs Fractint to seek out and identify a possible Finite Attractor
  and, if found, to display the Level Sets of its Basin of Attraction, in
  addition to those of the Basin of Attraction of Infinity.  In many cases
  this results in a "lake" with colored "waves" in it;  in other cases
  there may be little change in the lake's appearance.

  For a quick demonstration, select a fractal type of LAMBDA, with a
  parameter of 0.5 + 0.5i.  You will obtain an image with a large blue
  lake.  Now set "Look for finite attractor" to 1 with the "Y" menu.  The
  image will be re-drawn with a much more colorful lake.  A Finite
  Attractor lives in the center of one of the resulting "ripple" patterns
  in the lake - turn the <O>rbits display on to see where it is - the
  orbits of all initial points that are in the lake converge there.

  Fractint tests for the presence of a Finite Attractor by iterating a
  Critical Value of the fractal's function.  If the iteration doesn't bail
  out before exceeding twice the iteration limit, it is almost certain
  that we have a Finite Attractor - we assume that we have.

  Next we define a small circle around it and, after each iteration, as
  well as testing for the usual bailout value being exceeded, we test to
  see if we've hit the circle. If so, we bail out and color our pixels
  according to the number of iterations performed.  Result - a nicely
  colored-in lake that displays the Level Sets of the Basin of Attraction
  of the Finite Attractor.  Sometimes !

  First exception: This does not work for the lakes of Mandel-types.
  Every point in a Mandel-type is, in effect, a single point plucked from
  one of its related Julia-types.  A Mandel-type's lake has an infinite
  number of points, and thus an infinite number of related Julia-type
  sets, and consequently an infinite number of finite attractors too.  It
  *MAY* be possible to color in such a lake, by determining the attractor
  for EVERY pixel, but this would probably treble (at least) the number of
  iterations needed to draw the image.  Due to this overhead, Finite
  Attractor logic has not been implemented for Mandel-types.

  Secondly, certain Julia-types with lakes may not respond to this
  treatment, depending on the parameter value used.  E.g., the Lambda Set
  for 0.5 + 0.5i responds well; the Lambda Set for 0.0 + 1.0i does not -
  its lake stays blue.  Attractors that consist of single points, or a
  cycle of a finite number of points are ok.  Others are not.  If you're
  into fractal technospeak, the implemented approach fails if the Julia-
  type is a Parabolic case, or has Siegel Disks, or has Herman Rings.

  However, all the difficult cases have one thing in common - they all
  have a parameter value that falls exactly on the edge of the related
  Mandel-type's lake.  You can avoid them by intelligent use of the
  Mandel-Julia Space-Bar toggle:  Pick a view of the related Mandel-type
  where the center of the screen is inside the lake, but not too close to
  its edge, then use the space-bar toggle.  You should obtain a usable
  Julia-type with a lake, if you follow this guideline.

  Thirdly, the initial implementation only works for Julia-types that use
  the "Standard" fractal engine in Fractint.  Fractals with their own
  special algorithms are not affected by Finite Attractor logic, as yet.

                     Fractint Version xx.xx                     Page 154

  Finally, the finite attractor code will not work if it fails to detect a
  finite attractor.  If the number of iterations is set too low, the
  finite attractor may be missed.

  Despite these restrictions, the Finite Attractor logic can produce
  interesting results.  Just bear in mind that it is principally a bonus
  off-shoot from the development of the MAGNETic fractal types, and is not
  specifically tuned for optimal performance for other Julia types.

  (Thanks to Kevin Allen for the above).

  There is a second type of finite attractor coloring, which is selected
  by setting "Look for Finite Attractor" to a negative value.  This colors
  points by the phase of the convergence to the finite attractor, instead
  of by the speed of convergence.

  For example, consider the Julia set for -0.1 + 0.7i, which is the three-
  lobed "rabbit" set.  The Finite Attractor is an orbit of length three;
  call these values a, b, and c.  Then, the Julia set iteration can
  converge to one of three sequences: a,b,c,a,b,c,..., or b,c,a,b,c,...,
  or c,a,b,c,a,b,...  The Finite Attractor phase option colors the
  interior of the Julia set with three colors, depending on which of the
  three sequences the orbit converges to.  Internally, the code determines
  one point of the orbit, say "a", and the length of the orbit cycle, say
  3.  It then iterates until the sequence converges to a, and then uses
  the iteration number modulo 3 to determine the color.


  TRIG IDENTITIES

  The following trig identities are invaluable for coding fractals that
  use complex-valued transcendental functions of a complex variable in
  terms of real-valued functions of a real variable, which are usually
  found in compiler math libraries. In what follows, we sometimes use "*"
  for multiplication, but leave it out when clarity is not lost. We use
  "^" for exponentiation; x^y is x to the y power.

     (u+iv) + (x+iy) = (u+x) + i(v+y)
     (u+iv) - (x+iy) = (u-x) + i(v-y)
     (u+iv) * (x+iy) = (ux - vy) + i(vx + uy)
     (u+iv) / (x+iy) = ((ux + vy) + i(vx - uy)) / (x^2 + y^2)

     e^(x+iy)   = (e^x) (cos(y) + i sin(y))

     log(x+iy) = (1/2)log(x*x + y*y) + i(atan(y/x) + 2kPi)
        for k = 0, -1, 1, -2, 2, ...
         (Fractint generally uses only the principle value, k=0. The log
          function refers to log base e, or ln.)

     z^w = e^(w*log(z))

     sin(x+iy)  = sin(x)cosh(y) + i cos(x)sinh(y)
     cos(x+iy)  = cos(x)cosh(y) - i sin(x)sinh(y)
     tan(x+iy)  = sin(x+iy) / cos(x+iy)
     sinh(x+iy) = sinh(x)cos(y) + i cosh(x)sin(y)
     cosh(x+iy) = cosh(x)cos(y) + i sinh(x)sin(y)

                     Fractint Version xx.xx                     Page 155

     tanh(x+iy) = sinh(x+iy) / cosh(x+iy)
     cosxx(x+iy) = cos(x)cosh(y) + i sin(x)sinh(y)
       (cosxx is present in Fractint to provide compatibility with a bug
       which was in its cos calculation before version 16)

                       sin(2x)               sinh(2y)
     tan(x+iy) = ------------------  + i------------------
                 cos(2x) + cosh(2y)     cos(2x) + cosh(2y)

                   sin(2x) - i*sinh(2y)
     cotan(x+iy) = --------------------
                    cosh(2y) - cos(2x)

                      sinh(2x)                sin(2y)
     tanh(x+iy) = ------------------ + i------------------
                  cosh(2x) + cos(2y)    cosh(2x) + cos(2y)

                    sinh(2x) - i*sin(2y)
     cotanh(x+iy) = --------------------
                     cosh(2x) - cos(2y)

     asin(z) = -i * log(i*z+sqrt(1-z*z))
     acos(z) = -i * log(z+sqrt(z*z-1))
     atan(z) = i/2* log((1-i*z)/(1+i*z))

     asinh(z) = log(z+sqrt(z*z+1))
     acosh(z) = log(z+sqrt(z*z-1))
     atanh(z) = 1/2*log((1+z)/(1-z))

     sqr(x+iy) = (x^2-y^2) + i*2xy
     sqrt(x+iy) = sqrt(sqrt(x^2+y^2)) * (cos(atan(y/x)/2) + i sin(atan(y/x)/2))

     ident(x+iy) = x+iy
     conj(x+iy) = x-iy
     recip(x+iy) = (x-iy)/(x^2+y^2)
     flip(x+iy) = y+ix
     zero(x+iy) = 0
     cabs(x+iy) = sqrt(x^2 + y^2)

  Fractint's definitions of abs(x+iy) and |x+iy| below are non-standard.
  Math texts define both absolute value and modulus of a complex number to
  be the same thing.  They are both equal to cabs(x+iy) as defined above.

     |x+iy| = x^2 + y^2
     abs(x+iy) = sqrt(x^2) + i sqrt(y^2)

  Quaternions are four dimensional generalizations of complex numbers.
  They almost obey the familiar field properties of real numbers, but fail
  the commutative law of multiplication, since x*y is not generally equal
  to y*x.

  Quaternion algebra is most compactly described by specifying the rules
  for multiplying the basis vectors 1, i, j, and k. Quaternions form a
  superset of the complex numbers, and the basis vectors 1 and i are the
  familiar basis vectors for the complex algebra. Any quaternion q can be
  represented as a linear combination q = x + yi + zj + wk of the basis

                     Fractint Version xx.xx                     Page 156

  vectors just as any complex number can be written in the form z = a +
  bi.

  Multiplication rules for quaternion basis vectors:
  ij =  k jk =  i ki = j
  ji = -k kj = -i ik = -j
  ii = jj = kk = -1
  ijk = -1

  Note that ij = k but ji = -k, showing the failure of the commutative law.
  The rules for multiplying any two quaternions follow from the behavior
  of the basis vectors just described. However, for your convenience, the
  following formula works out the details.

  Let q1 = x1 + y1i + z1j + w1k and q2 = x2 + y2i + z2j + w2k.
  Then q1q2 = 1(x1x2 - y1y2 - z1z2 - w1w2) +
              i(y1x2 + x1y2 + w1z2 - z1w2) +
              j(z1x2 - w1y2 + x1z2 + y1w2) +
              k(w1x2 + z1y2 - y1z2 + x1w2)

  Quaternions are not the only possible four dimensional supersets of the
  complex numbers. William Hamilton, the discoverer of quaternions in the
  1830's, considered the alternative called the hypercomplex number
  system.  Unlike quaternions, the hypercomplex numbers satisfy the
  commutative law of multiplication. The law which fails is the field
  property that states that all non-zero elements of a field have a
  multiplicative inverse. For a non-zero hypercomplex number h, the
  multiplicative inverse 1/h does not always exist.

  As with quaternions, we will define multiplication in terms of the basis
  vectors 1, i, j, and k, but with subtly different rules.

  Multiplication rules for hypercomplex basis vectors:
  ij = k  jk = -i ki = -j
  ji = k  kj = -i ik = -j
  ii = jj = -kk = -1
  ijk = 1

  Note that now ij = k and ji = k, and similarly for other products of pairs
  of basis vectors, so the commutative law holds.

  Hypercomplex multiplication formula:
  Let h1 = x1 + y1i + z1j + w1k and h2 = x2 + y2i + z2j + w2k.
  Then  h1h2 =  1(x1x2 - y1y2 - z1z2 + w1w2) +
                i(y1x2 + x1y2 - w1z2 - z1w2) +
                j(z1x2 - w1y2 + x1z2 - y1w2) +
                k(w1x2 + z1y2 + y1z2 + x1w2)

  As an added bonus, we'll give you the formula for the reciprocal.

  Let det = [((x-w)^2+(y+z)^2)((x+w)^2+(y-z)^2)]
  Then 1/h =   1[ x(x^2+y^2+z^2+w^2)-2w(xw-yz)]/det +
               i[-y(x^2+y^2+z^2+w^2)-2z(xw-yz)]/det +
               j[-z(x^2+y^2+z^2+w^2)-2y(xw-yz)]/det +
               k[ w(x^2+y^2+z^2+w^2)-2x(xw-yz)]/det

                     Fractint Version xx.xx                     Page 157

  A look at this formula shows the difficulty with hypercomplex numbers.
  In order to calculate 1/h, you have to divide by the quantity det =
  [((x-w)^2+(y+z)^2)((x+w)^2+(y-z)^2)]. So when this quantity is zero, the
  multiplicative inverse will not exist.

  Hypercomplex numbers numbers have an elegant generalization of any unary
  complex valued function defined on the complex numbers. First, note that
  hypercomplex numbers can be represented as a pair of complex numbers in
  the following way.
  Let h = x + yi + zj + wk.
      a = (x-w) + i(y+z)
      b = (x+w) + i(y-z)
  The numbers a and b are complex numbers. We can represent h as the pair
  of complex numbers (a,b). Conversely, if we have a hypercomplex number
  given to us in the form (a,b), we can solve for x, y, z, and w. The
  solution to
     c = (x-w) + i(y+z)
     d = (x+w) + i(y-z)
  is
     x = (real(c) + real(d))/2
     y = (imag(c) + imag(d))/2
     z = (imag(c) - imag(d))/2
     x = (real(d) - real(c))/2
  We can now, for example, define sin(h) as (sin(a),sin(b)). We know how
  to compute sin(a) and sin(b) (see trig identities above).

  Let c = sin(a) and d = sin(b). Now use the equations above to solve for
  x, y, z, and w in terms of c and d. The beauty of this is that it really
  doesn't make any difference what function we use. Instead of sin, we
  could have used cos, sinh, ln, or z^2. Using this technique, Fractint
  can create 3-D fractals using the formula h' = fn(h) + c, where "fn" is
  any of the built-in functions. Where fn is sqr(), this is the famous
  mandelbrot formula, generalized to four dimensions.

  For more information, see _Fractal Creations, Second Edition_ by Tim
  Wegner and Bert Tyler, Waite Group Press, 1993.

                     Fractint Version xx.xx                     Page 158

 Appendix B Stone Soup With Pixels: The Authors

  THE STONE SOUP STORY

  Once upon a time, somewhere in Eastern Europe, there was a great famine.
  People jealously hoarded whatever food they could find, hiding it even
  from their friends and neighbors. One day a peddler drove his wagon into
  a village, sold a few of his wares, and began asking questions as if he
  planned to stay for the night.

  [No!  No!  It was three Russian Soldiers! - Lee Crocker]
  [Wait!  I heard it was a Wandering Confessor! - Doug Quinn]
  [Well *my* kids have a book that uses Russian Soldiers! - Bert]
  [Look, who's writing this documentation, anyway? - Monte]
  [Ah, but who gets it *last* and gets to upload it? - Bert]

  "There's not a bite to eat in the whole province," he was told. "Better
  keep moving on."

  "Oh, I have everything I need," he said. "In fact, I was thinking of
  making some stone soup to share with all of you." He pulled an iron
  cauldron from his wagon, filled it with water, and built a fire under
  it.  Then, with great ceremony, he drew an ordinary-looking stone from a
  velvet bag and dropped it into the water.

  By now, hearing the rumor of food, most of the villagers had come to the
  square or watched from their windows. As the peddler sniffed the "broth"
  and licked his lips in anticipation, hunger began to overcome their
  skepticism.

  "Ahh," the peddler said to himself rather loudly, "I do like a tasty
  stone soup. Of course, stone soup with CABBAGE -- that's hard to beat."

  Soon a villager approached hesitantly, holding a cabbage he'd retrieved
  from its hiding place, and added it to the pot. "Capital!" cried the
  peddler. "You know, I once had stone soup with cabbage and a bit of salt
  beef as well, and it was fit for a king."

  The village butcher managed to find some salt beef...and so it went,
  through potatoes, onions, carrots, mushrooms, and so on, until there was
  indeed a delicious meal for all. The villagers offered the peddler a
  great deal of money for the magic stone, but he refused to sell and
  traveled on the next day. And from that time on, long after the famine
  had ended, they reminisced about the finest soup they'd ever had.

                                  ***

  That's the way Fractint has grown, with quite a bit of magic, although
  without the element of deception. (You don't have to deceive programmers
  to make them think that hours of painstaking, often frustrating work is
  fun... they do it to themselves.)

  It wouldn't have happened, of course, without Benoit Mandelbrot and the
  explosion of interest in fractal graphics that has grown from his work
  at IBM. Or without the example of other Mandelplotters for the PC. Or
  without those wizards who first realized you could perform Mandelbrot

                     Fractint Version xx.xx                     Page 159

  calculations using integer math (it wasn't us - we just recognize good
  algorithms when we steal--uhh--see them).  Or those graphics experts who
  hang around the CompuServe PICS forum and keep adding video modes to the
  program.  Or...

  A WORD ABOUT THE AUTHORS

  Fractint is the result of a synergy between the main authors, many
  contributors, and published sources.  All four of the main authors have
  had a hand in many aspects of the code.  However, each author has
  certain areas of greater contribution and creativity.  Since there is
  not room in the credits screen for the contributions of the main
  authors, we list these here to facilitate those who would like to
  communicate with us on particular subjects.

  Main Authors of Version 19.

  BERT TYLER is the original author of Fractint.  He wrote the "blindingly
  fast" 386-specific 32 bit integer math code and the original video mode
  logic. Bert made Stone Soup possible, and provides a sense of direction
  when we need it. His forte is writing fast 80x86 assembler, his
  knowledge of a variety of video hardware, and his skill at hacking up
  the code we send him!

  Bert has a BA in mathematics from Cornell University.  He has been in
  programming since he got a job at the computer center in his sophomore
  year at college - in other words, he hasn't done an honest day's work in
  his life.  He has been known to pass himself off as a PC expert, a UNIX
  expert, a statistician, and even a financial modeling expert.  He is
  currently masquerading as an independent PC consultant, supporting the
  PC-to-Mainframe communications environment at NIH.  If you sent mail
  from the Internet to an NIH staffer on his 3+Mail system, it was
  probably Bert's code that mangled it during the Internet-to-3+Mail
  conversion.  He also claims to support the MS-Kermit environment at NIH.
  Fractint is Bert's first effort at building a graphics program.

  TIM WEGNER contributed the original implementation of palette animation,
  and is responsible for most of the 3D mechanisms.  He provided the main
  outlines of the "StandardFractal" engine and data structures, and is
  accused by his cohorts of being "obsessed with options". One of Tim's
  main interests is the use of four dimensional algebras to produce
  fractals.  Tim served as team coordinator for version 19, and integrated
  Wes Loewer's arbitrary precision library into Fractint.

  Tim has BA and MA degrees in mathematics from Carleton College and the
  University of California Berkeley.  He worked for 7 years overseas as a
  volunteer, doing things like working with Egyptian villagers building
  water systems. Since returning to the US in 1982, he has written shuttle
  navigation software, a software support environment prototype, and
  supported strategic information planning, all at NASA's Johnson Space
  Center. Tim has started his own business, and now writes and programs
  full time.

  JONATHAN OSUCH started throwing pebbles into the soup around version
  15.0 with a method for simulating an if-then-else structure using the
  formula parser.  He has contributed the fn||fn fractal types, the built-

                     Fractint Version xx.xx                     Page 160

  in bailout tests, the increase in both the maximum iteration count and
  bailout value, and bug fixes too numerous to count. Jonathan worked
  closely with Robin Bussell to implement Robin's browser mechanism in
  Fractint.

  Jonathan has a B.S. in Physics from the University of Dubuque and a B.S.
  in Computer Science from Mount Mercy College, both in Iowa.  He is
  currently working as a consultant in the nuclear power industry.

  WES LOEWER first got his foot in the Stone Soup door by writing fast
  floating point assembler routines for Mandelbrot, Julia, and Lyapunov
  fractals.  He also rewrote the boundary trace algorithms and added the
  frothybasin fractal.  His most significant contribution is the addition
  of the arbitrary precision library which allows Fractint to perform
  incredibly deep zooms.

  Wes has a B.S. in Physics from Wheaton College in Illinois.  He also
  holds an M.S. in Physics and an M.Ed. in Education from Texas A&M
  University.  Wes teaches physics and math at McCullough High School in
  The Woodlands, Texas where his pupils inspire him to keep that sense of
  amazement that students get when they understand a physical or
  mathematical principle for the first time.  Since he uses Fractint to
  help teach certain mathematical principles, he's one of the few folks
  who actually gets to use Fractint on the job.  Besides his involvement
  with Fractint, Wes is the author of WL-Plot, an equation graphing
  program, and MatCalc, a matrix calculator program.

  DISTRIBUTION OF FRACTINT

  New versions of FRACTINT are uploaded to the CompuServe network, and
  make their way to other systems from that point.  FRACTINT is available
  as two self-extracting archive files - FRAINT.EXE (executable &
  documentation) and FRASRC.EXE (source code).

  The latest version can always be found in one of CompuServe's GO
  GRAPHICS forums. Alas, the GO GRAPHICS Group is growing so fast that we
  get moved around from periodically, and rumor has it that yet another
  move is imminent.  The current location of Fractint is the "Fractal
  Sources" library of the GO GRAPHDEV forum. The forum staff will leave
  pointers to our new home if we are moved again.

  If you're not a CompuServe subscriber, but wish to get more information
  about CompuServe and its graphics forums, feel free to call their 800
  number (800-848-8199) and ask for operator number 229.

  If you don't have access to CompuServe, many other sites tend to carry
  these files shortly after their initial release (although sometimes
  using different naming conventions).  For instance...

  If you speak Internet and FTP, SIMTEL20 and its various mirror sites
  tend to carry new versions of Fractint shortly after they are released.
  look in the /SimTel/msdos/graphics directory for files named FRA*.*.
  Then again, if you don't speak Internet and FTP...

                     Fractint Version xx.xx                     Page 161

  Your favorite local BBS probably carries these files as well (although
  perhaps not the latest versions) using naming conventions like FRA*.ZIP.
  One BBS that *does* carry the latest version is the "Ideal Studies BBS"
  (508)757-1806, 1200/2400/9600HST.  Peter Longo is the SYSOP and a true
  fractal fanatic.  There is a very short registration, and thereafter the
  entire board is open to callers on the first call.  Then again, if you
  don't even have a modem...

  Many Shareware/Freeware library services will ship you diskettes
  containing the latest versions of Fractint for a nominal fee that
  basically covers their cost of packaging and a small profit that we
  don't mind them making.  One in particular is the Public (Software)
  Library, PO Box 35705, Houston, TX 77235-5705, USA.  Their phone number
  is 800-242-4775 (outside the US, dial 713-524-6394).  Ask for item #9112
  for five 5.25" disks, #9113 for three 3.5" disks.  Cost is $6.99 plus $4
  S&H in the U.S./Canada, $11 S&H overseas.

  In Europe, the latest versions are available from another Fractint
  enthusiast, Jon Horner - Editor of FRAC'Cetera, a disk-based
  fractal/chaos resource.  Disk prices for UK/Europe are: 5.25" HD
  BP4.00/4.50  : 3.5" HD BP (British Pounds) 4.00/4.50.  Prices include
  p&p (airmail to Europe).  Contact: Jon Horner, FRAC'Cetera, Le Mont
  Ardaine, Rue des Ardaines, St. Peters, Guernsey GY7 9EU, CI, UK.  Phone
  (44) 01481 63689.  CIS 100112,1700

  The X Windows port of Fractint maintained by Ken Shirriff is available
  via FTP from ftp.cs.berkeley.edu in /ucb/sprite/xfract.

  CONTACTING THE AUTHORS
  Communication between the authors for development of the next version of
  Fractint takes place in a CompuServe (CIS) GO GRAPHICS GROUP (GGG)
  forum.  This forum changes from time to time as as the GGG grows. You
  can always find it using the CompuServe GO GRAPHICS command. Currently
  we are located in GRAPHDEV (Graphics Developers) forum, Section 4
  (Fractal Sources).

  Most of the authors have never met except on CompuServe. Access to the
  GRAPHDEV forum is open to any and all interested in computer generated
  fractals. New members are always welcome! Stop on by if you have any
  questions or just want to take a peek at what's getting tossed into the
  soup.  This is by far the best way to have your questions answered or
  participate in discussion. Also, you'll find many GIF image files
  generated by fellow Fractint fans and many fractal programs as well in
  the GRAPHDEV forum's data library 5.

  If you're not a CompuServe subscriber, but wish to get more information
  about CompuServe and its graphics forums, feel free to call their 800
  number (800-848-8199) and ask for operator number 229.

  The following authors have agreed to the distribution of their
  addresses.  Usenet/Internet/Bitnet/Whatevernet users can reach CIS users
  directly if they know the user ID (i.e., Bert Tyler's ID is
  73477.433@compuserve.com).

                     Fractint Version xx.xx                     Page 162

  Just remember that CIS charges by the minute, so it costs us a little
  bit to read a message -- don't kill us with kindness. And don't send all
  your mail to Bert -- spread it around a little! Postal addresses are
  listed below so that you have a way to send bug reports and ideas to the
  Stone Soup team.

  Please understand that we receive a lot of mail, and because of the
  demands of volunteer work on Fractint as well as our professional
  responsibilities, we are generally unable to answer it all.  Several of
  us have reached the point where we can't answer any conventional mail.
  We *do* read and enjoy all the mail we receive, however. If you need a
  reply, the best thing to do is use email, which we are generally able to
  answer, or better yet, leave a message in CompuServe's GRAPHDEV.

  (This address list is getting seriously out of date. We have updated
  information from those folks who have contacted us. The next release of
  Fractint will contain the addresses of *only* those people who have
  explicitly told us that their address is correct and they want it
  listed. Please contact one of the main authors with this information.)


                     Fractint Version xx.xx                     Page 163

  Current main authors:

  Bert Tyler              [73477,433] on CIS
  Tyler Software          (which is also 73477.433@compuserve.com, if
  124 Wooded Lane          you're on the Internet - see above)
  Villanova, PA 19085
  (610) 525-5478

  Timothy Wegner          [71320,675] on CIS
  4714 Rockwood           twegner@phoenix.net (Internet)
  Houston, TX 77004
  (713) 747-7543

  Jonathan Osuch          [73277,1432] on CIS
  2110 Northview Drive
  Marion, IA  52302

  Wesley Loewer           loewer@tenet.edu on INTERNET
  78 S. Circlewood Glen
  The Woodlands, TX  77381
  (713) 292-3449

  Contributing authors' addresses (in alphabetic order).

  Joseph A Albrecht
  9250 Old Cedar Ave #215
  Bloomington, Mn 55425
  (612) 884-3286

  Kevin C Allen           kevina@microsoft.com on Internet
  9 Bowen Place
  Seven Hills
  NSW 2147
  Australia
  +61-2-870-2297 (Work)
  (02) 831-4821 (Home)

  Rob Beyer               [71021,2074] on CIS
  23 Briarwood Lane
  Laguna Hills, CA, 92656
  (714) 957-0227
  (7-12pm PST & weekends)

  John W. Bridges         (Author GRASP/Pictor, Imagetools, PICEM, VGAKIT)
  2810 Serang Place Costa Mesa
  California 92626-4827   [75300,2137] on CIS, GENIE:JBRIDGES

  Juan J Buhler           jbuhler@usina.org.ar
  Santa Fe 2227 1P "E"
  (54-1) 84 3528
  Buenos Aires, Argentina

  Michael D. Burkey       burkey@sun9.math.utk.edu on Internet
  6600 Crossgate Rd.
  Knoxville, TN 37912

                     Fractint Version xx.xx                     Page 164

  Robin Bussell
  13 Bayswater Rd
  Horfield
  Bristol
  Avon, England
  (044)-0272-514451

  Prof Jm Collard-Richard jmc@math.ethz.ch

  Monte Davis             [71450,3542] on CIS
  223 Vose Avenue
  South Orange, NJ 07079
  (201) 378-3327

  Paul de Leeuw
  50 Henry Street
  Five Dock
  New South Wales
  2046
  Australia
  +61-2-396-2246 (Work)
  +61-2-713-6064 (Home)

  David Guenther          [70531,3525] on CIS
  50 Rockview Drive
  Irvine, CA 92715

  Michael L. Kaufman      kaufman@eecs.nwu.edu on INTERNET
  2247 Ridge Ave, #2K     (also accessible via EXEC-PC bbs)
  Evanston, IL, 60201
  (708) 864-7916
  Joe McLain              [75066,1257] on CIS
  McLain Imaging
  2417 Venier
  Costa Mesa, CA 92627
  (714) 642-5219

  Bob Montgomery          [73357,3140] on CIS
  (Author of VPIC)
  132 Parsons Road
  Longwood, Fl  32779

  Roy Murphy              [76376,721] on CIS
  9050 Ewing Ave.
  Evanston, IL 60203

  Ethan Nagel             [71062,3677] on CIS
  4209 San Pedro NE #308
  Albuquerque, NM 87109
  (505) 884-7442
  Mark Peterson           [73642,1775] on CIS
  The Yankee Programmer
  405-C Queen St., Suite #181
  Southington, CT 06489
  (203) 276-9721

                     Fractint Version xx.xx                     Page 165

  Marc Reinig             [72410,77] on CIS
  3415 Merrill Rd.        72410.77@compuserve.com.
  Aptos, CA. 95003
  (408) 475-2132

  Lee H. Skinner          [75450,3631] on CIS
  P.O. Box 14944
  Albuquerque, NM  87191
  (505) 293-5723

  Dean Souleles           [75115,1671] on CIS
  8840 Collett Ave.
  Sepulveda, CA  91343
  (818) 893-7558

  Chris J Lusby Taylor
  32 Turnpike Road
  Newbury, England
  Tel 011 44 635 33270

  Scott Taylor                  [72401,410] on CIS
  2913 Somerville Drive Apt #1  scott@bohemia.metronet.org on Internet
  Ft. Collins, Co  80526        DGWM18A on Prodigy
  (303) 221-1206

  Paul Varner             [73237,441] on CIS
  PO Box 930
  Shepherdstown, WV 25443
  (304) 876-2011

  Phil Wilson             [76247,3145] on CIS
  410 State St., #55
  Brooklyn, NY 11217
  (718) 624-5272

                     Fractint Version xx.xx                     Page 166

 Appendix C GIF Save File Format

  Since version 5.0, Fractint has had the <S>ave-to-disk command, which
  stores screen images in the extremely compact, flexible .GIF (Graphics
  Interchange Format) widely supported on CompuServe. Version 7.0 added
  the <R>estore-from-disk capability.

  Until version 14, Fractint saved images as .FRA files, which were a non-
  standard extension of the then-current GIF87a specification.  The reason
  was that GIF87a did not offer a place to store the extra information
  needed by Fractint to implement the <R> feature -- i.e., the parameters
  that let you keep zooming, etc.  as if the restored file had just been
  created in this session.  The .FRA format worked with all of the popular
  GIF decoders that we tested, but these were not true GIF files. For one
  thing, information after the GIF terminator (which is where we put the
  extra info) has the potential to confuse the online GIF viewers used on
  CompuServe. For another, it is the opinion of some GIF developers that
  the addition of this extra information violates the GIF87a spec. That's
  why we used the default filetype .FRA instead.

  Since version 14, Fractint has used a genuine .GIF format, using the
  GIF89a spec - an upwardly compatible extension of GIF87a, released by
  CompuServe on August 1 1990.  This new spec allows the placement of
  application data within "extension blocks".  In version 14 we changed
  our default savename extension from .FRA to .GIF.

  There is one significant advantage to the new GIF89a format compared to
  the old GIF87a-based .FRA format for Fractint purposes:  the new .GIF
  files may be uploaded to the CompuServe graphics forums fractal
  information intact.  Therefore anyone downloading a Fractint image from
  CompuServe will also be downloading all the information needed to
  regenerate the image.

  Fractint can still read .FRA files generated by earlier versions.  If
  for some reason you wish to save files in the older GIF87a format, for
  example because your favorite GIF decoder has not yet been upgraded to
  GIF89a, use the command-line parameter "GIF87a=yes".  Then any saved
  files will use the original GIF87a format without any application-
  specific information.

  An easy way to convert an older .FRA file into true .GIF format suitable
  for uploading is something like this at the DOS prompt:
      FRACTINT MYFILE.FRA SAVENAME=MYFILE.GIF BATCH=YES
  Fractint will load MYFILE.FRA, save it in true .GIF format as
  MYFILE.GIF, and return to DOS.

  GIF and "Graphics Interchange Format" are trademarks of CompuServe
  Incorporated, an H&R Block Company.

                     Fractint Version xx.xx                     Page 167

 Appendix D Other Fractal Products

  (Forgive us, but we just *have* to begin this section with a plug for
  *our* fractal products...)

  Several of Fractint's programmers have written books about fractals,
  Fractint, and Winfract (the Windows version of Fractint).  The book
  about Fractint is Fractal Creations Second Edition (1994 Waite Group
  Press, ISBN # 1-878739-34-4).  The book about Winfract is The Waite
  Group's Fractals for Windows (1992 Waite Group Press, ISBN # 1-878739-
  25-5).

  Fractal Creations Second Edition includes:
   o A guided tour of Fractint.
   o A detailed manual and reference section of commands.
   o A tutorial on fractals.
   o A reference containing tips, explanations, and examples of parameters
     for all the Fractals generated by Fractint/Winfract.
   o Secrets on how the programs work internally.
   o Spectacular color plate section.
   o A CD containing Fractint and Xfract source and executable, and over a
     thousand spectacular fractal images.
   o A complete copy of the source code with a chapter explaining how the
     program works.

  If you enjoy Fractint, you're sure to enjoy Fractal Creations. The book
  includes Fractint and is an excellent companion to the program.  If you
  use the Windows environment, be sure to pick up a copy of Fractals for
  Windows as well.

  A great fractals newsletter is "Amygdala" published by Rollo Silver.
  You'll find equal parts fractal algorithms, humor, reviews, and ideas.
  Write to:
     Amygdala
     Box 219
     San Cristobal, NM 87564
     USA
     Email:rsilver@lanl.gov
     Phone: 505-586-0197

  Another great fractals newsletter (this one based in the UK) is
  "FRAC'Cetera", a disk-based fractal/chaos resource, for PCs and
  compatibles, distributed on 3.5" HD disk, published by Jon Horner.
  Contact:

     Jon Horner
     FRAC'Cetera
     Le Mont Ardaine
     Rue des Ardaines
     ST Peters
     Guernsey GY7 9EU, CI, UK
     Email: 100112.1700@compuserve.com
     PH: (44) 01481 63689

                     Fractint Version xx.xx                     Page 168

  Several Fractint enthusiasts are selling Fractal CDs.  Two of the best
  are called "Fractal Frenzy" by Lee Skinner, and "Fractography" by Caren
  Park.  Highly recommended original artwork in a variety of graphics
  formats.

  You can receive the "Fractal Frenzy CD" by sending $39.95US + $5.00 S&H
     to Walnut Creek CDROM
     1537 Palos Verdes Mall, Suite 260
     Walnut Creek, CA 94596

  and the "Fractography" CD by sending $30.00US + $5.00 S&H (in US/Canada)
  to
     Lost and Found Books
     485 Front Street N, Suite A
     Issaquah, WA 98027-2900

  Michael Peters (author of PARTOBAT) and Randall Scott have written a
  fractal program called HOP based on the Martin orbit fractals. This
  program is much narrower than Fractint in the kind of thing that it
  does, but has many more animation effects and makes a great screen
  saver. Michael sent us the algorithms for the chip, quadruptwo, and
  threeply fractal types to give us a taste. The file is called HOPZIP.EXE
  in LIB 4 of CompuServe's GRAPHDEV forum.

                     Fractint Version xx.xx                     Page 169

 Appendix E Bibliography

  BARNSLEY, Michael: "Fractals Everywhere," Academic Press, 1988.

  DAVENPORT, Clyde: "A Hypercomplex Calculus with Applications to
     Relativity", ISBN 0-9623837-0-8. This self-published expansion of Mr.
     Davenport's Master's thesis makes the case for using hypercomplex
     numbers rather than quaternions. This book provided the background
     for Fractint's implementation of hypercomplex fractals.

  DEWDNEY, A. K., "Computer Recreations" columns in "Scientific American"
     -- 8/85, 7/87, 11/87, 12/88, 7/89.

  FEDER, Jens: "Fractals," Plenum, 1988.
     Quite technical, with good coverage of applications in fluid
     percolation, game theory, and other areas.

  GLEICK, James: "Chaos: Making a New Science," Viking Press, 1987.
     The best non-technical account of the revolution in our understanding
     of dynamical systems and its connections with fractal geometry.

  MANDELBROT, Benoit: "The Fractal Geometry of Nature," W. H. Freeman &
     Co., 1982.
     An even more revised and expanded version of the 1977 work. A rich
     and sometimes confusing stew of formal and informal mathematics, the
     prehistory of fractal geometry, and everything else. Best taken in
     small doses.

  MANDELBROT, Benoit: "Fractals: Form, Chance, and Dimension," W. H.
     Freeman & Co., 1977.
     A much revised translation of "Les objets fractals: forme, hasard, et
     dimension," Flammarion, 1975.

  PEITGEN, Heinz-Otto & RICHTER, Peter: "The Beauty of Fractals,"
     Springer-Verlag, 1986.
     THE coffee-table book of fractal images, knowledgeable on computer
     graphics as well as the mathematics they portray.

  PEITGEN, Heinz-Otto & SAUPE, Ditmar: "The Science of Fractal Images,"
     Springer-Verlag, 1988.
     A fantastic work, with a few nice pictures, but mostly filled with
     *equations*!!!

  PICKOVER, Clifford: "Computers, Pattern, Chaos, and Beauty," St.
     Martin's Press, 1990.

  SCHROEDER, Manfred: "Fractals, Chaos, Power Laws," W. H. Freeman & Co.,
     1991.

  WEGNER, Timothy: "Image Lab, Second Edition", Waite Group Press, to be
     released in 1995. Learn how to create fractal animations, fractal RDS
     stereo images, and how to use Fractint with other image creation and
     processing tools such as Piclab, POV-Ray and Polyray ray tracers.

                     Fractint Version xx.xx                     Page 170

  WEGNER, Timothy & TYLER, Bert: "Fractal Creations, Second Edition" Waite
     Group Press, 1993
     This is the definitive Fractint book. Spectacular color plate
     section, totally new and expanded fractal type descriptions,
     annotated PAR files, source code secrets, and a CD filled to the brim
     with spectacular fractals.

  WEGNER, Timothy, TYLER, Bert, PETERSON, Mark, and Branderhorst, Pieter:
     "Fractals for Windows," Waite Group Press, 1992.
     This book is to Winfract (the Windows version of Fractint) what
     "Fractal Creations" is to Fractint.
  .

                     Fractint Version xx.xx                     Page 171

 Appendix F Other Programs

  WINFRACT. Bert Tyler has ported Fractint to run under Windows 3!  The
  same underlying code is used, with a Windows user interface.  Winfract
  has almost all the functionality of Fractint - the biggest difference is
  the absence of a zillion weird video modes.  Fractint for DOS will
  continue to be the definitive version.  Winfract is available from
  CompuServe in GRAPHDEV Lib 4, as WINFRA.ZIP (executable) and WINSRC.ZIP
  (source).


  PICLAB, by Lee Crocker - a freeware image manipulation utility available
  from CompuServe in PICS Lib 10, as PICLAB.EXE.  PICLAB can do very
  sophisticated resizing and color manipulation of GIF and TGA files.  It
  can be used to reduce 24 bit TGA files generated with the Fractint
  "lightname" option to GIF files.


  FDESIGN, by Doug Nelson (CIS ID 70431,3374) - a freeware IFS fractal
  generator available from CompuServe in GRAPHDEV Lib 4, and probably on
  your local BBS.  This program requires a VGA adapter and a Microsoft-
  compatible mouse, and a floating point coprocessor is highly
  recommended.  It generates IFS fractals in a *much* more intuitive
  fashion than Fractint.  It can also (beginning with version 3.0) save
  its IFS formulas in Fractint-style .IFS files.

  ACROSPIN, by David Parker - An inexpensive commercial program that reads
  an object definition file and creates images that can be rapidly rotated
  in three dimensions. The Fractint "orbitsave=yes" option creates files
  that this program can read for orbit-type fractals and IFS fractals.
  Contact:
     David Parker                         801-966-2580
     P O Box 26871                        800-227-6248
     Salt Lake City, UT  84126-0871

                     Fractint Version xx.xx                     Page 172

 Appendix G Revision History

  Versions 18.1 and 18.2 are bug-fix releases for version 18.0.  Changes
  from 18.1 to 18.2 include:

   The <b> command now causes filenames only to be written in PAR files.

   Fractint will now search directories in the PATH for files not found in
   the requested the requested directory or the current directory. If you
   place .MAP, .FRM, etc. in directories in your PATH, then Fractint will
   find them.

   Fixed bug that caused fractals using PI symmetry to fail at high
   resolution.

   Fractals interrupted with <3> or <r> can now resume.

   The palette editor's <u> (undo) now works.

   The <s> command in orbit/Julia window mode is no longer case sensitive.

   Added warnings that the POV-Ray output is obsolete (but has been left
   in).  Use POV-Ray's height field facility instead or create and convert
   RAW files.

   Fixed several IFS bugs.

  Changes from 18.0 to 18.1 include:

   Overlay tuning - the Mandelbrot/Julia Set fractals are now back up to
   17.x speeds

   Disk Video modes now work correctly with VESA video adapters (they used
   to use the same array for different purposes, confusing each other)

   1024x768x256 and 2048x2048x256 disk video modes work again

   Parameter-file processing no longer crashes Fractint if it attempts to
   run a formula requiring access to a non-existent FRM file

   IFS arrays no longer overrun their array space

   type=cellular fixes

   "autologmap=2" now correctly picks up the minimum color

   The use of disk-video mode with random-access fractal types is now
   legal (it generates a warning message but lets you proceed if you
   really want to)

   The Lsystems "spinning-wheel" now spins slower (removing needless
   overhead)

   Changes to contributors' addresses in the Help screens

                     Fractint Version xx.xx                     Page 173

  (The remainder of this "new features" section is from version 18.0)

  New fractal types:

   19 new fractal types, including:

   New fractal types - 'lambda(fn||fn)', 'julia(fn||fn)',
   'manlam(fn||fn)', 'mandel(fn||fn)', 'halley', 'phoenix', 'mandphoenix',
   'cellular', generalized bifurcation, and 'bifmay' - from Jonathan
   Osuch.

   New Mandelcloud, Quaternion, Dynamic System, Cellular Automata fractal
   types from Ken Shirriff.

   New HyperComplex fractal types from Timothy Wegner

   New ICON type from Dan Farmer, including a PAR file of examples.

   New Frothy Basin fractal types (and PAR entries) by Wesley Loewer

   MIIM (Modified Inverse Iteration Method) implementation of Inverse
   Julia from Michael Snyder.

   New Inverse Julia fractal type from Juan Buhler.

   New floating-point versions of Markslambda, Marksmandel, Mandel4, and
   Julia4 types (chosen automatically if the floating-point option is
   enabled).

  New options/features:

   New assembler-based parser logic from Chuck Ebbert - significantly
   faster than the C-based code it replaces!

   New assembler-based Lyapunov logic from Nicholas Wilt and Wes Loewer.
   Roughly six times faster than the old version!

   New Orbits-on-a-window / Julia-in-a-window options:
    1) The old Overlay option is now '#' (Shift-3).
    2) During generation, 'O' brings up orbits (as before) - after
       generation, 'O' brings up new orbits Windows mode.
    3) Control-O brings up new orbits Windows mode at any time.
    4) Spacebar toggles between Inverse Julia mode and the Julia set and
       back to the Mandelbrot set.
   These new "in-a-window" modes are really neat!  See Orbits Window
   (p. 26) and Julia Toggle Spacebar Commands (p. 35) for details.

   New multi-image GIF support in the <B> command.  You can now generate
   65535x65535x256 fractal images using Fractint (if you have the disk
   space, of course).  This option builds special PAR entries and a
   MAKEMIG.BAT file that you later use to invoke Fractint multiple times
   to generate individual sections of the image and (in a final step)
   stitch them all together.  If your other software can't handle
   Multiple-image GIFs, a SIMPLGIF program is also supplied that converts
   MIGS into simgle-image GIFs.  Press F1 at the <B> prompts screen for
   details.

                     Fractint Version xx.xx                     Page 174

   Fractint's decoder now handles Multi-Image Gifs.

   New SuperVGA/VESA Autodetect logic from the latest version of VGAKIT.
   Sure hope we didn't break anything.

   New register-compatible 8514/A code from Jonathan Osuch.  By default,
   Fractint now looks first for the presence of an 8514/A register-
   compatible adapter and then (and only if it doesn't find one) the
   presence of the 8514/A API (IE, HDILOAD is no longer necessary for
   register-compatible "8514/a" adapters).  Fractint can be forced to use
   the 8514/A API by using a new command-line option, "afi=yes".  Jonathan
   also added ATI's "8514/a-style" 800x600x256 and 1280x1024x16 modes.

   New XGA-detection logic for ISA-based XGA-2 systems.

   The palette editor now has a "freestyle" editing option.  See Palette
   Editing Commands (p. 18) for details.


   Fractint is now more "batch file" friendly.  When running Fractint from
   a batch file, pressing any key will cause Fractint to exit with an
   errorlevel = 2.  Any error that interrupts an image save to disk will
   cause an exit with errorlevel = 2.  Any error that prevents an image
   from being generated will cause an exit with errorlevel = 1.

   New Control-X, Control-Y, and Control-Z options flip a fractal image
   along the X-axis, Y-axis, and Origin, respectively.

   New area calculation mode in TAB screen from Ken Shirriff (for accuracy
   use inside=0).

   The TAB screen now indicates when the Integer Math algorithms are in
   use.

   The palette must now be explicitly changed, it will not reset to the
   default unexpectedly when doing things like switching video modes.

   The Julibrot type has been generalized.  Julibrot fractals can now be
   generated from PAR files.

   Added <b> command support for viewwindows.

   Added room for two additional PAR comments in the <B> command

   New coloring method for IFS shows which parts of fractal came from
   which transform.

   Added attractor basin phase plotting for Julia sets from Ken Shirriff.

   Improved finite attractor code to find more attractors from Ken
   Shirriff.

   New zero function, to be used in PAR files to replace old integer tan,
   tanh

                     Fractint Version xx.xx                     Page 175

   Debugflag=10000 now reports video chipset in use as well as CPU/FPU
   type and available memory

   Added 6 additional parameters for params= for those fractal types that
   need them.

   New 'matherr()' logic lets Fractint get more aggressive when these
   errors happen.

   New autologmap option (log=+-2) from Robin Bussell that ensures that
   all palette values are used by searching the screen border for the
   lowest value and then setting log= to +- that color.

   Two new diffusion options - falling and square cavity.

   Three new Editpal commands: '!', '@' and '#' commands (that's <shift-
   1>, <shift-2>, and <shift-3>) to swap R<->G, G<->B, R<->B.

   Parameter files now use a slightly shorter maximum line length, making
   them a bit more readable when stuffed into messages on CompuServe.

   Plasma now has 16-bit .POT output for use with Ray tracers. The "old"
   algorithm has been modified so that the plasma effect is independent of
   resolution.

   Slight modification to the Raytrace code to make it compatible with
   Rayshade 4.0 patch level 6.

   Improved boundary-tracing logic from Wesley Loewer.

   Command-line parameters can now be entered on-the-fly using the <g> key
   thanks to Ken Shirriff.

   Dithered gif images can now be loaded onto a b/w display.  Thanks to
   Ken Shirriff.

   Pictures can now be output as compressed PostScript.  Thanks to Ken
   Shirriff.

   Periodicity is a new inside coloring option.  Thanks to Ken Shirriff.

   Fixes: symmetry values for the SQR functions, bailout for the floating-
   pt versions of 'lambdafn' and 'mandelfn' fractals from Jonathan Osuch.

   "Flip", "conj" operators are now selectable in the parser

   New DXF Raytracing option from Dennis Bragg.

   Improved boundary-tracing logic from Wesley Loewer.

   New MSC7-style overlay structure is used if MAKEFRAC.BAT specifies
   MSC7.  (with new FRACTINT.DEF and FRACTINT.LNK files for MSC7 users).
   Several modules have been re-organized to take advantage of this new
   overlay capability if compiled under MSC7.

                     Fractint Version xx.xx                     Page 176

   Fractint now looks first any embedded help inside FRACTINT.EXE, and
   then for an external FRACTINT.HLP file before giving up. Previous
   releases required that the help text be embedded inside FRACTINT.EXE.

  Bug fixes:

   Corrected formulas displayed for Marksmandel, Cmplxmarksmandel, and
    associated julia types.

   BTM and precision fixes.

   Symmetry logic changed for various "outside=" options

   Symmetry value for EXP function in lambdafn and lambda(fn||fn) fixed.

   Fixed bug where math errors prevented save in batch mode.

   The <3> and <r> commands no longer destroy image -- user can back out
   with ESC and image is still there.

   Fixed display of correct number of Julibrot parameters, and Julibrot
   relaxes and doesn't constantly force ALTERN.MAP.

   Fixed tesseral type for condition when border is all one color but
   center contains image.

   Fixed integer mandel and julia when used with parameters > +1.99 and <
   -1.99

   Eliminated recalculation when generating a julia type from a mandelbrot
   type when the 'z' screen is viewed for the first time.

   Minor logic change to prevent double-clutching into and out of graphics
   mode when pressing, say, the 'x' key from a menu screen.

   Changed non-US phone number for the Houston Public (Software) Library

   The "Y" screen is now "Extended Options" instead of "Extended Doodads"

   ...and probably a lot more bux-fixes that we've since forgotten that
   we've implemented.


  Version 17.2, 3/92

   - Fixed a bug which caused Fractint to hang when a Continuous Potential
     Bailout value was set (using the 'Y') screen and then the 'Z' screen
     was activated.
   - fixed a bug which caused "batch=yes" runs to abort whenever any
     key was pressed.
   - bug-fixes in the Stereo3D/Targa logic from Marc Reinig.
   - Fractint now works correctly again on FPU-less 8088s when
     zoomed deeply into the Mandelbrot/Julia sets
   - The current image is no longer marked as "not resumable" on a
     Shell-To-Dos ("D") command.
   - fixed a bug which prevented the "help" functions from working

                     Fractint Version xx.xx                     Page 177

     properly during fractal-type selection for some fractal types.

  Version 17.1, 3/92

   - fixed a bug which caused PCs with no FPU to lock up when they
   attempted
     to use some fractal types.
   - fixed a color-cycling bug which caused the palette to single-step
     when you pressed ESCAPE to exit color-cycling.
   - fixed the action of the '<' and '>' keys during color-cycling.

  Version 17.0, 2/92

  - New fractal types (but of course!):

  Lyapunov Fractals from Roy Murphy (see Lyapunov Fractals (p. 62) for
  details)

  'BifStewart' (Stewart Map bifurcation) fractal type and new bifurcation
  parameters (filter cycles, seed population) from Kevin Allen.

  Lorenz3d1, Lorenz3d3, and Lorenz3d4 fractal types from Scott Taylor.
  Note that a bug in the Lorenz3d1 fractal prevents zooming-out from
  working with it at the moment.

  Martin, Circle, and Hopalong (culled from Dewdney's Scientific American
  Article)

  Lots of new entries in fractint.par.

  New ".L" files (TILING.L, PENROSE.L)

  New 'rand()' function added to the 'type=formula' parser

  - New fractal generation options:

  New 'Tesseral' calculation algorithm (use the 'X' option list to select
  it) from  Chris Lusby Taylor.

  New 'Fillcolor=' option shows off Boundary Tracing and Tesseral
  structure

  inside=epscross and inside=startrail options taken from a paper by
  Kenneth Hooper, with credit also to Clifford Pickover

  New Color Postscript Printer support from Scott Taylor.

  Sound= command now works with <O>rbits and <R>ead commands.

  New 'orbitdelay' option in X-screen and command-line interface

  New "showdot=nn" command-line option that displays the pixel currently
  being worked on using the specified color value (useful for those
  lloooonngg images being calculated using solid guessing - "where is it
  now?").

                     Fractint Version xx.xx                     Page 178

  New 'exitnoask=yes' commandline/SSTOOLS.INI option to avoid the final
  "are you sure?" screen

  New plasma-cloud options.  The interface at the moment (documented here
    and here only because it might change later) lets you:
    - use an alternate drawing algorithm that gives you an earlier preview
      of the finished image.  - re-generate your favorite plasma cloud
    (say, at a higher resolution)
      by forcing a re-select of the random seed.

  New 'N' (negative palette) option from Scott Taylor - the documentation
  at this point is:  Pressing 'N' while in the palette editor will invert
  each color. It will convert only the current color if it is in 'x' mode,
  a range if in 'y' mode, and every color if not in either the 'x' or 'y'
  mode.

  - Speedups:

  New, faster floating-point Mandelbrot/Julia set code from Wesley Loewer,
  Frank Fussenegger and Chris Lusby Taylor (in separate contributions).

  Faster non-386 integer Mandelbrot code from Chris Lusby Taylor, Mike
  Gelvin and Bill Townsend (in separate contributions)

  New integer Lsystems logic from Nicholas Wilt

  Finite-Attractor fixups and Lambda/mandellambda speedups from Kevin
  Allen.

  GIF Decoder speedups from Mike Gelvin

  - Bug-fixes and other enhancements:

  Fractint now works with 8088-based AMSTRAD computers.

  The video logic is improved so that (we think) fewer video boards will
  need "textsafe=save" for correct operation.

  Fixed a bug in the VESA interface which effectively messed up adapters
  with unusual VESA-style access, such as STB's S3 chipset.

  Fixed a color-cycling bug that would at times restore the wrong colors
  to your image if you exited out of color-cycling, displayed a 'help'
  screen, and then returned to the image.

  Fixed the XGA video logic so that its 256-color modes use the same
  default 256 colors as the VGA adapter's 320x200x256 mode.

  Fixed the 3D bug that caused bright spots on surfaces to show as black
  blotches of color 0 when using a light source.

  Fixed an image-generation bug that sometimes caused image regeneration
  to restart even if not required if the image had been zoomed in to the
  point that floating-point had been automatically activated.

                     Fractint Version xx.xx                     Page 179

  Added autodetection and 640x480x256 support for the Compaq Advanced VGA
  Systems board - I wonder if it works?

  Added VGA register-compatible 320x240x256 video mode.

  Fixed the "logmap=yes" option to (again) take effect for continuous
  potential images.  This was broken in version 15.x.

  The colors for the floating-point algorithm of the Julia fractal now
  match the colors for the integer algorithm.

  If the GIF Encoder (the "Save" command) runs out of disk space, it now
  tells you about it.

  If you select both the boundary-tracing algorithm and either "inside=0"
  or "outside=0", the algorithm will now give you an error message instead
  of silently failing.

  Updated 3D logic from Marc Reinig.

  Minor changes to permit IFS3D fractal types to be handled properly using
  the "B" command.

  Minor changes to the "Obtaining the latest Source" section to refer to
  BBS access (Peter Longo's) and mailed diskettes (the Public (Software)
  Library).


  Version 16.12, 8/91

    Fix to cure some video problems reported with Amstrad 8088/8086-based
       PCs.

  Version 16.11, 7/91

    SuperVGA Autodetect fixed for older Tseng 3000 adapters.

    New "adapter=" options to force the selection of specific SuperVGA
       adapter types.  See Video Parameters (p. 111) for details.

    Integer/Floating-Point math toggle is changed only temporarily if
       floating-point math is forced due to deep zooming.

    Fractint now survives being modified by McAfee's "SCAN /AV" option.

    Bug Fixes for Acrospin interface, 3D "Light Source Before
       Transformation" fill type, and GIF decoder.

    New options in the <Z> parameters screen allow you to directly enter
       image coordinates.

    New "inside=zmag" and "outside=real|imag|mult|summ" options.

    The GIF Decoder now survives reading GIF files with a local color map.
    Improved IIT Math Coprocessor support.

                     Fractint Version xx.xx                     Page 180

    New color-cycling single-step options, '<' and '>'.

  Version 16.0, 6/91

    Integrated online help / fractint.doc system from Ethan Nagel.  To
      create a printable fractint.doc file see Startup Parameters (p. 101)
      .

    Over 350 screens of online help! Try pressing <F1> just about
    anywhere!

    New "autokey" feature.  Type "demo" to run the included demo.bat and
      demo.key files for a great demonstration of Fractint.  See Autokey
      Mode (p. 72) for details.

    New <@> command executes a saved set of commands.  The <b> command has
      changed to write the current image's parameters as a named set of
      commands in a structured file.  Saved sets of commands can
      subsequently be executed with the <@> command.  See Parameter
      Save/Restore Commands (p. 23).  A default "fractint.par" file is
      included with the release.

    New <z> command allows changing fractal type-specific parameters
      without going back through the <t> (fractal type selection) screen.

    Ray tracer interface from Marc Reinig, generates 3d transform output
      for a number of ray tracers; see "Interfacing with Ray Tracing
      Programs" (p. 96)

    Selection of video modes and structure of "fractint.cfg" have changed.
      If you have a customized fractint.cfg file, you'll have to rebuild
      it based on this release's version. You can customize the assignment
      of your favorite video modes to function keys; see Video Mode
      Function Keys (p. 28).  <delete> is a new command key which goes
      directly to video mode selection.

    New "cyclerange" option (command line and <y> options screen) from
      Hugh Steele. Limits color cycling to a specified range of colors.

    Improved Distance Estimator Method (p. 74) algorithm from Phil
    Wilson.

    New "ranges=" option from Norman Hills.  See Logarithmic Palettes and
      Color Ranges (p. 77) for details.

    type=formula definitions can use "variable functions" to select sin,
      cos, sinh, cosh, exp, log, etc at run time; new built-ins tan, tanh,
      cotan, cotanh, and flip are available with type=formula; see Type
      Formula (p. 55)

    New <w> command in palette editing mode to convert image to greyscale

    All "fn" fractal types (e.g. fn*fn) can now use new functions tan,
      tanh, cotan, cotanh, recip, and ident; bug in prior cos function
      fixed, new function cosxx (conjugate of cos) is the old erroneous
      cos calculation

                     Fractint Version xx.xx                     Page 181

    New L-Systems from Herb Savage
    New IFS types from Alex Matulich
    Many new formulas in fractint.frm, including a large group from JM
      Collard-Richard
    Generalized type manzpwr with complex exponent per Lee Skinner's
    request
    Initial orbit parameter added to Gingerbreadman fractal type

    New color maps (neon, royal, volcano, blues, headache) from Daniel
    Egnor

    IFS type has changed to use a single file containing named entries
      (instead of a separate xxx.ifs file per type); the <z> command
      brings up IFS editor (used to be <i> command).  See Barnsley IFS
      Fractals (p. 43).

    Much improved support for PaintJet printers; see PaintJet Parameters
    (p. 117)

    From Scott Taylor:
      Support for plotters using HP-GL; see Plotter Parameters (p. 118)
      Lots of new PostScript halftones; see PostScript Parameters (p. 115)
      "printer=PS[L]/0/..." for full page PostScript; see PostScript
      Parameters (p. 115)
      Option to drive printer ports directly (faster); see Printer
      Parameters (p. 114)
      Option to change printer end of line control chars; see Printer
      Parameters (p. 114)

    Support for XGA video adapter
    Support for Targa+ video adapter
    16 color VGA mode enhancements:
      Now use the first 16 colors of .map files to be more predictable
      Palette editor now works with these modes
      Color cycling now works properly with these modes Targa video
    adapter fixes; Fractint now uses (and requires) the "targa"
      and "targaset" environment variables for Targa systems
    "vesadetect=no" parameter to bypass use of VESA video driver; try
      this if you encounter video problems with a VESA driver Upgraded
    video adapter detect and handling from John Bridges; autodetect
      added for NCR, Trident 8900, Tseng 4000, Genoa (this code is from a
      beta release of VGAKIT, we're not sure it all works yet)

    Zoom box is included in saved/printed images (but, is not recognized
      as anything special when such an image is restored)

    The colors numbers reserved by the palette editor are now selectable
      with the new <v> palette editing mode command

    Option to use IIT floating point chip's special matrix arithmetic for
      faster 3D transforms; see "fpu=" in Startup Parameters (p. 101)

    Disk video cache increased to 64k; disk video does less seeking when
      running to real disk
    Faster floating point code for 287 and higher fpus, for types mandel,
      julia, barnsleyj1/m1/j2/m2, lambda, manowar, from Chuck Ebbert

                     Fractint Version xx.xx                     Page 182

    "filename=.xxx" can be used to set default <r> function file mask

    Selection of type formula or lsys now goes directly to entry selection
      (file selection step is now skipped); to change to a different file,
      use <F6> from the entry selection screen

    Three new values have been added to the textcolors= parameter; if you
      use this parameter you should update it by inserting values for the
      new 6th, 7th, 9th, and 13th positions; see "textcolors=" in Color
      Parameters (p. 106)

    The formula type's imag() function has changed to return the result as
      a real number

    Fractal type-specific parameters (entered after selecting a new
      fractal type with <T>) now restart at their default values each time
      you select a new fractal type

    Floating point input fields can now be entered in scientific notation
      (e.g.  11.234e-20). Entering the letters "e" and "p" in the first
      column causes the numbers e=2.71828... and pi=3.14159... to be
      entered.

    New option "orbitsave=yes" to create files for Acrospin for some types
      (see Barnsley IFS Fractals (p. 43), Orbit Fractals (p. 50),
      Acrospin (p. 171))

    Bug fixes:
      Problem with Hercules adapter auto-detection repaired.
      Problems with VESA video adapters repaired (we're not sure we've got
        them all yet...)
      3D transforms fixed to work at high resolutions (> 1000 dots).
      3D parameters no longer clobbered when restoring non-3D images.
      L-Systems fixed to not crash when order too high for available
      memory.
      PostScript EPS file fixes.
      Bad leftmost pixels with floating point at 2048 dot resolution
      fixed.
      3D transforms fixed to use current <x> screen float/integer setting.
      Restore of images using inversion fixed.
      Error in "cos" function (used with "fn" type fractals) fixed; prior
        incorrect function still available as "cosxx" for compatibility

    Old 3D=nn/nn/nn/... form of 3D transform parameters no longer
    supported

    Fractint source code now Microsoft C6.00A compatible.

  Version 15.11, 3/91, companion to Fractal Creations, not for general
  release

    Autokey feature, IIT fpu support, and some bug fixes publicly released
    in version 16.

                     Fractint Version xx.xx                     Page 183

  Version 15 and 15.1, 12/90

    New user interface! Enjoy! Some key assignments have changed and some
      have been removed.
    New palette editing from Ethan Nagel.
    Reduced memory requirements - Fractint now uses overlays and will run
      on a 512K machine.
    New <v>iew command: use to get small window for fast preview, or to
      setup an image which will eventually be rendered on hard copy with
      different aspect ratio
    L-System fractal type from Adrian Mariano
    Postscript printer support from Scott Taylor
    Better Tandy video support and faster CGA video from Joseph A Albrecht
    16 bit continuous potential files have changed considerably;  see the
      Continuous Potential section for details.  Continuous potential is
      now resumable.
    Mandelbrot calculation is faster again (thanks to Mike Gelvin) -
      double speed in 8086 32 bit case
    Compressed log palette and sqrt palette from Chuck Ebbert
    Calculation automatically resumes whenever current image is resumable
      and is not paused for a visible reason.
    Auto increment of savename changed to be more predictable
    New video modes:
      trident 1024x768x256 mode
      320x480x256 tweak mode (good for reduced 640x480 viewing)
      changed NEC GB-1, hopefully it works now
    Integer mandelbrot and julia now work with periodicitycheck
    Initial zoombox color auto-picked for better contrast (usually)
    New adapter=cga|ega|mcga|vga for systems having trouble with auto-
    detect
    New textsafe=no|yes for systems having trouble with garbled text mode
    <r> and <3> commands now present list of video modes to pick from; <r>
      can reduce a non-standard or unviewable image size.
    Diffusion fractal type is now resumable after interrupt/save
    Exitmode=n parameter, sets video mode to n when exiting from fractint
    When savetime is used with 1 and 2 pass and solid guessing, saves are
      deferred till the beginning of a new row, so that no calculation
      time is lost.
    3d photographer's mode now allows the first image to be saved to disk
    textcolors=mono|12/34/56/... -- allows setting user interface colors
    Code (again!) compilable under TC++ (we think!)
    .TIW files (from v9.3) are no longer supported as input to 3D
      transformations
    bug fixes:
      multiple restores (msc 6.0, fixed in 14.0r)
      repeating 3d loads problem; slow 3d loads of images with float=yes
      map= is now a real substitute for default colors
      starfield and julibrot no longer cause permanent color map
      replacement
      starfield parameters bug fix - if you couldn't get the starfield
      parameters to do anything interesting before, try again with this
      release
      Newton and newtbasin orbit display fixed

                     Fractint Version xx.xx                     Page 184

   Version 15.1:

    Fixed startup and text screen problems on systems with VESA compliant
      video adapters.
    New textsafe=save|bios options.
    Fixes for EGA with monochrome monitor, and for Hercules Graphics Card.
      Both should now be auto-detected and operate correctly in text
      modes.  Options adapter=egamono and adapter=hgc added.
    Fixed color L-Systems to not use color 0 (black).
    PostScript printing fix.

  Version 14, 8/90

    LAST MINUTE NEWS FLASH!
      CompuServe announces the GIF89a on August 1, 1990, and Fractint
      supports it on August 2! GIF files can now contain fractal
      information!  Fractint now saves its files in the new GIF89a format
      by default, and uses .GIF rather than .FRA as a default filetype.
      Note that Fractint still *looks* for a .FRA file on file restores if
      it can't find a .GIF file, and can be coerced into using the old
      GIF87a format with the new 'gif87a=yes' command-line option.

    Pieter Branderhorst mounted a major campaign to get his name in
    lights:
    Mouse interface:  Diagonals, faster movement, improved feel. Mouse
      button assignments have changed - see the online help.
    Zoom box enhancements:  The zoom box can be rotated, stretched,
      skewed, and panned partially offscreen.  See "More Zoom Box
      Commands".
    FINALLY!! You asked for it and we (eventually, by talking Pieter into
      it [actually he grabbed it]) did it!  Images can be saved before
      completion, for a subsequent restore and continue.  See
      "Interrupting and Resuming" and "Batch Mode".
    Off-center symmetry:  Fractint now takes advantage of x or y axis
      symmetry anywhere on the screen to reduce drawing time.
    Panning:  If you move an image up, down, left, or right, and don't
      change anything else, only the new edges are calculated.
    Disk-video caching - it is now possible, reasonable even, to do most
      things with disk video, including solid guessing, 3d, and plasma.
    Logarithmic palette changed to use all colors.  It now matches regular
      palette except near the "lake".  "logmap=old" gets the old way.
    New "savetime=nnn" parameter to save checkpoints during long
    calculations.
    Calculation time is shown in <Tab> display.

    Kevin C Allen    Finite Attractor, Bifurcation Engine, Magnetic
    fractals...
    Made Bifurcation/Verhulst into a generalized Fractal Engine (like
      StandardFractal, but for Bifurcation types), and implemented
      periodicity checking for Bifurcation types to speed them up.
    Added Integer version of Verhulst Bifurcation (lots faster now).
      Integer is the default.  The Floating-Point toggle works, too.
    Added NEW Fractal types BIFLAMBDA, BIF+SINPI, and BIF=SINPI. These are
      Bifurcation types that make use of the new Engine. Floating-
      point/Integer toggle is available for BIFLAMBDA. The SINPI types are
      Floating-Point only, at this time.

                     Fractint Version xx.xx                     Page 185

    Corrected the generation of the MandelLambda Set.  Sorry, but it's
      always been wrong (up to v 12, at least).  Ask Mandelbrot !
    Added NEW Fractal types MAGNET1M, MAGNET1J, MAGNET2M, MAGNET2J from
      "The Beauty of Fractals".  Floating-Point only, so far, but what do
      you expect with THESE formulae ?!
    Added new symmetry types XAXIS NOIMAG and XAXIS NOREAL, required by
      the new MAGNETic Fractal types.
    Added Finite Attractor Bailout (FAB) logic to detect when iterations
      are approaching a known finite attractor. This is required by the
      new MAGNETic Fractal types.
    Added Finite Attractor Detection (FAD) logic which can be used by
      *SOME* Julia types prior to generating an image, to test for finite
      attractors, and find their values, for use by FAB logic. Can be used
      by the new MAGNETic Fractal Types, Lambda Sets, and some other Julia
      types too.

    Mike Burkey sent us new tweaked video modes:
      VGA     - 400x600x256   376x564x256   400x564x256
      ATI VGA - 832x612x256 New HP Paintjet support from Chris Martin
    New "FUNCTION=" command to allow substition of different
      transcendental functions for variables in types (allows one type
      with four of these variables to represent 7*7*7*7 different types!
    ALL KINDS of new fractal types, some using "FUNCTION=": fn(z*z),
      fn*fn, fn*z+z, fn+fn, sqr(1/fn), sqr(fn), spider, tetrate, and
      Manowar. Most of these are generalizations of formula fractal types
      contributed by Scott Taylor and Lee Skinner.
    Distance Estimator logic can now be applied to many fractal types
      using distest= option. The types "demm" and "demj" have been
      replaced by "type=mandel distest=nnn" and "type=julia distest=nnn"
    Added extended memory support for diskvideo thanks to Paul Varner
    Added support for "center and magnification" format for corners.
    Color 0 is no longer generated except when specifically requested with
      inside= or outside=.
    Formula name is now included in <Tab> display and in <S>aved images.
    Bug fixes - formula type and diskvideo, batch file outside=-1 problem.
    Now you can produce your favorite fractal terrains in full color
      instead of boring old monochrome! Use the fullcolor option in 3d!
      Along with a few new 3D options.
    New "INITORBIT=" command to allow alternate Mandelbrot set orbit
      initialization.


  Version 13.0, 5/90

    F1 was made the help key.
      Use F1 for help
      Use F9 for EGA 320x200x16 video mode
      Use CF4 for EGA 640x200x16 mode (if anybody uses that mode)
    Super-Solid-guessing (three or more passes) from Pieter Branderhorst
      (replaces the old solid-guessing mode)
    Boundary Tracing option from David Guenther ("fractint passes=btm", or
      use the new 'x' options screen)
    "outside=nnn" option sets all points not "inside" the fractal to color
      "nnn" (and generates a two-color image).
    'x' option from the main menu brings up a full-screen menu of many
      popular options and toggle switches

                     Fractint Version xx.xx                     Page 186

    "Speed Key" feature for fractal type selection (either use the cursor
      keys for point-and-shoot, or just start typing the name of your
      favorite fractal type)
    "Attractor" fractals (Henon, Rossler, Pickover, Gingerbread)
    Diffusion fractal type by Adrian Mariano
    "type=formula" formulas from Scott Taylor and Lee H. Skinner.
    "sound=" options for attractor fractals.  Sound=x  plays speaker tones
      according to the 'x' attractor value  Sound=y  plays speaker tones
      according to the 'y' attractor value.  Sound=z  plays speaker tones
      according to the 'z' attractor value  (These options are best
      invoked with the floating-point algorithm flag set.)
    "hertz=" option for adjusting the "sound=x/y/z" output.
    Printer support for color printers (printer=color) from Kurt Sowa
    Trident 4000 and Oak Technologies SuperVGA support from John Bridges
    Improved 8514/A support (the zoom-box keeps up with the cursor keys
    now!)
    Tandy 1000 640x200x16 mode from Brian Corbino (which does not, as yet,
      work with the F1(help) and TAB functions)
    The Julibrot fractal type and the Starmap option now automatically
      verify that they have been selected with a 256-color palette, and
      search for, and use, the appropriate GLASSESn.MAP or ALTERN.MAP
      palette map when invoked.  *You* were supposed to be doing that
      manually all along, but *you* probably never read the docs, huh?
    Bug Fixes:
      TAB key now works after R(estore) commands
      PS/2 Model 30 (MCGA) adapters should be able to select 320x200x256
        mode again (we think)
      Everex video adapters should work with the Autodetect modes again
        (we think)


  Version 12.0, 3/90

    New SuperVGA Autodetecting and VESA Video modes (you tell us the
      resolution you want, and we'll figure out how to do it)
    New Full-Screen Entry for most prompting
    New Fractal formula interpreter ('type=formula') - roll your own
      fractals without using a "C" compiler!
    New 'Julibrot' fractal type
    Added floating point option to all remaining fractal types.
    Real (funny glasses) 3D - Now with "real-time" lorenz3D!!
    Non-Destructive <TAB> - Check out what your fractal parameters are
      without stopping the generation of a fractal image
    New Cross-Hair mode for changing individual palette colors (VGA only)
    Zooming beyond the limits of Integer algorithms (with automatic
      switchover to a floating-point algorithm when you zoom in "too far")
    New 'inside=bof60', 'inside=bof61' ("Beauty of Fractals, Page nn")
    options
    New starmap ('a' - for astrology? astronomy?) transformation option
    Restrictions on the options available when using Expanded Memory
      "Disk/RAM" video mode have been removed
    And a lot of other nice little clean-up features that we've already
      forgotten that we've added...
    Added capability to create 3D projection images (just barely) for
      people with 2 or 4 color video boards.

                     Fractint Version xx.xx                     Page 187

  Version 11.0, 1/90

    More fractal types
      mandelsinh/lambdasinh        mandelcosh/lambdacosh
      mansinzsqrd/julsinzsqrd      mansinexp/julsinexp
      manzzprw/julzzpwr            manzpower/julzpower
      lorenz (from Rob Beyer)      lorenz3d
      complexnewton                complexbasin
      dynamic                      popcorn
    Most fractal types given an integer and a floating point algorithm.
      "Float=yes" option now determines whether integer or floating-point
      algorithms are used for most fractal types.  "F" command toggles the
      use of floating-point algorithms, flagged in the <Tab> status
      display
    8/16/32/../256-Way decomposition option (from Richard Finegold)
    "Biomorph=", "bailout=", "symmetry="  and "askvideo=" options
    "T(ransform)" option in the IFS editor lets you select 3D options
      (used with the Lorenz3D fractal type)
    The "T(ype)" command uses a new "Point-and-Shoot" method of selecting
      fractal types rather than prompting you for a type name
    Bug fixes to continuous-potential algorithm on integer fractals, GIF
      encoder, and IFS editor


  Version 10.0, 11/89

    Barnsley IFS type (Rob Beyer)
    Barnsley IFS3D type
    MandelSine/Cos/Exp type
    MandelLambda/MarksLambda/Unity type
    BarnsleyM1/J1/M2/J2/M3/J3 type
    Mandel4/Julia4 type
    Sierpinski gasket type
    Demm/Demj and bifurcation types (Phil Wilson), "test" is "mandel"
    again
    <I>nversion command for most fractal types
    <Q>uaternary decomposition toggle and "DECOMP=" argument
    <E>ditor for Barnsley IFS parameters
    Command-line options for 3D parameters
    Spherical 3D calculations 5x faster
    3D now clips properly to screen edges and works at extreme perspective
    "RSEED=" argument for reproducible plasma clouds
    Faster plasma clouds (by 40% on a 386)
    Sensitivity to "continuous potential" algorithm for all types except
      plasma and IFS
    Palette-map <S>ave and Restore (<M>) commands
    <L>ogarithmic and <N>ormal palette-mapping commands and arguments
    Maxiter increased to 32,000 to support log palette maps
    .MAP and .IFS files can now reside anywhere along the DOS path
    Direct-video support for Hercules adapters (Dean Souleles)
    Tandy 1000 160x200x16 mode (Tom Price)
    320x400x256 register-compatible-VGA "tweaked" mode
    ATI VGA Wonder 1024x768x16 direct-video mode (Mark Peterson)
    1024x768x16 direct-video mode for all supported chipsets
    Tseng 640x400x256 mode
    "Roll-your-own" video mode 19

                     Fractint Version xx.xx                     Page 188

    New video-table "hot-keys" eliminate need for enhanced keyboard to
      access later entries


  Version 9.3, 8/89

    <P>rint command and "PRINTER=" argument (Matt Saucier)
    8514/A video modes (Kyle Powell)
    SSTOOLS.INI sensitivity and '@THISFILE' argument
    Continuous-potential algorithm for Mandelbrot/Julia sets
    Light source 3D option for all fractal types
    "Distance estimator" M/J method (Phil Wilson) implemented as "test"
    type
    LambdaCosine and LambdaExponent types
    Color cycling mode for 640x350x16 EGA adapters
    Plasma clouds for 16-color and 4-color video modes
    Improved TARGA support (Joe McLain)
    CGA modes now use direct-video read/writes
    Tandy 1000 320x200x16 and 640x200x4 modes (Tom Price)
    TRIDENT chip-set super-VGA video modes (Lew Ramsey)
    Direct-access video modes for TRIDENT, Chips & Technologies, and ATI
      VGA WONDER adapters (John Bridges). and, unlike version 9.1, they
      WORK in version 9.3!)
    "zoom-out" (<Ctrl><Enter>) command
    <D>os command for shelling out
    2/4/16-color Disk/RAM video mode capability and 2-color video modes
      supporting full-page printer graphics
    "INSIDE=-1" option (treated dynamically as "INSIDE=maxiter")
    Improved <H>elp and sound routines (even a "SOUND=off" argument)
    Turbo-C and TASM compatibility (really!  Would we lie to you?)


  Version 8.1, 6/89

    <3>D restore-from-disk and 3D <O>verlay commands, "3D=" argument
    Fast Newton algorithm including inversion option (Lee Crocker)
    16-bit Mandelbrot/Julia logic for 386-class speed with non-386 PCs on
      "large" images (Mark Peterson)
    Restore now loads .GIF files (as plasma clouds)
    TARGA video modes and color-map file options (Joe McLain)
    30 new color-cycling palette options (<Shft><F1> to <Alt><F10>)
    "Disk-video, RAM-video, EMS-video" modes
    Lambda sets now use integer math (with 80386 speedups)
    "WARN=yes" argument to prevent over-writing old .GIF files


  Version 7.0, 4/89

    Restore from disk (from prior save-to-disk using v. 7.0 or later)
    New types: Newton, Lambda, Mandelfp, Juliafp, Plasma, Lambdasine
    Many new color-cycling options (for VGA adapters only)
    New periodicity logic (Mark Peterson)
    Initial displays recognize (and use) symmetry
    Solid-guessing option (now the default)
    Context-sensitive <H>elp
    Customizable video mode configuration file (FRACTINT.CFG)

                     Fractint Version xx.xx                     Page 189

    "Batch mode" option
    Improved super-VGA support (with direct video read/writes)
    Non-standard 360 x 480 x 256 color mode on a STANDARD IBM VGA!


  Version 6.0, 2/89

    32-bit integer math emulated for non-386 processors; FRACT386 renamed
      FRACTINT
    More video modes


  Version 5.1, 1/89

    Save to disk
    New! Improved! (and Incompatible!) optional arguments format
    "Correct" initial image aspect ratio
    More video modes


  Version 4.0, 12/88

    Mouse support (Mike Kaufman)
    Dynamic iteration limits
    Color cycling
    Dual-pass mode
    More video modes, including "tweaked" modes for IBM VGA and register-
      compatible adapters


  Version 3.1, 11/88

    Julia sets


  Version 2.1, 10/23/88 (the "debut" on CIS)

    Video table
    CPU type detector


  Version 2.0, 10/10/88

    Zoom and pan


  Version 1.0, 9/88

    The original, blindingly fast, 386-specific 32-bit integer algorithm

                     Fractint Version xx.xx                     Page 190

 Appendix H Version13 to Version 14 Type Mapping

  A number of types in Fractint version 13 and earlier were generalized in
  version 14. We added a "backward compatibility" hook that (hopefully)
  automatically translates these to the new form when the old files are
  read. Files may be converted via:

     FRACTINT OLDFILE.FRA SAVENAME=NEWFILE.GIF BATCH=YES

  In a few cases the biomorph flag was incorrectly set in older files.  In
  that case, add "biomorph=no" to the command line.

  This procedure can also be used to convert any *.fra file to the new
  GIF89a spec, which now allows storage of fractal information.


  TYPES CHANGED FROM VERSION 13 -


  V13 NAME                V14 NAME + PARAMETERS
  --------                --------------------------------------

  LOGMAP=YES              LOGMAP=OLD   for identical Logmap type

  DEMJ                    JULIA DISTEST=nnn

  DEMM                    MANDEL DISTEST=nnn

                          Note: DISTEST also available on many other types

  MANSINEXP               MANFN+EXP FUNCTION=SIN

                          Note: New functions for this type are
                                cos sinh cosh exp log sqr

  JULSINEXP               JULFN+EXP FUNCTION=SIN

                          Note: New functions for this type are
                                cos sinh cosh exp log sqr

  MANSINZSQRD             MANFN+ZSQRD FUNCTION=SQR/SIN

                          Note: New functions for this type are
                                cos sinh cosh exp log sqr

  JULSINZSQRD             JULFN+ZSQRD FUNCTION=SQR/SIN

                          Note: New functions for this type are
                                cos sinh cosh exp log sqr

  LAMBDACOS               LAMBDAFN FUNCTION=COS

  LAMBDACOSH              LAMBDAFN FUNCTION=COSH

                     Fractint Version xx.xx                     Page 191

  LAMBDAEXP               LAMBDAFN FUNCTION=EXP

  LAMBDASINE              LAMBDAFN FUNCTION=SIN

  LAMBDASINH              LAMBDAFN FUNCTION=SINH

                          Note: New functions for this type are
                                log sqr

  MANDELCOS               MANDELFN FUNCTION=COS

  MANDELCOSH              MANDELFN FUNCTION=COSH

  MANDELEXP               MANDELFN FUNCTION=EXP

  MANDELSINE              MANDELFN FUNCTION=SIN

  MANDELSINH              MANDELFN FUNCTION=SINH

                          Note: New functions for this type are
                                log sqr

  MANDELLAMBDA            MANDELLAMBDA INITORBIT=PIXEL

  POPCORN SYMMETRY=NONE   POPCORNJUL

  -------------------------------------------------------------

  Formulas from FRACTINT.FRM in version 13

  MANDELGLASS             MANDELLAMBDA INITORBIT=.5/0

  INVMANDEL               V13 divide bug may cause some image differences.

  NEWTON4                 V13 divide bug may cause some image differences.

  SPIDER                  V13 divide bug may cause some image differences.

  MANDELSINE              MANDELFN FUNCTION=SIN BAILOUT=50

  MANDELCOSINE            MANDELFN FUNCTION=COS BAILOUT=50

  MANDELHYPSINE           MANDELFN FUNCTION=SINH BAILOUT=50

  MANDELHYPCOSINE         MANDELFN FUNCTION=COSH BAILOUT=50

  SCOTTSIN PARAMS=nnn     FN+FN FUNCTION=SIN/SQR BAILOUT=nnn+3

  SCOTTSINH PARAMS=nnn    FN+FN FUNCTION=SINH/SQR BAILOUT=nnn+3

  SCOTTCOS PARAMS=nnn     FN+FN FUNCTION=COS/SQR BAILOUT=nnn+3

  SCOTTCOSH PARAMS=nnn    FN+FN FUNCTION=COSH/SQR BAILOUT=nnn+3

                     Fractint Version xx.xx                     Page 192

  SCOTTLPC PARAMS=nnn     FN+FN FUNCTION=LOG/COS BAILOUT=nnn+3

  SCOTTLPS PARAMS=nnn     FN+FN FUNCTION=LOG/SIN BAILOUT=nnn+3
                          Note: New functions for this type are
                          sin/sin sin/cos sin/sinh sin/cosh sin/exp
                          cos/cos cos/sinh cos/cosh cos/exp
                          sinh/sinh sinh/cosh sinh/exp sinh/log
                          cosh/cosh cosh/exp cosh/log
                          exp/exp exp/log exp/sqr log/log log/sqr sqr/sqr

  SCOTTSZSA PARAMS=nnn    FN(Z*Z) FUNCTION=SIN BAILOUT=nnn+3

  SCOTTCZSA PARAMS=nnn    FN(Z*Z) FUNCTION=COS BAILOUT=nnn+3

                          Note: New functions for this type are
                          sinh cosh exp log sqr

  SCOTTZSZZ PARAMS=nnn    FN*Z+Z FUNCTION=SIN BAILOUT=nnn+3

  SCOTTZCZZ PARAMS=nnn    FN*Z+Z FUNCTION=COS BAILOUT=nnn+3

                          Note: New functions for this type are
                          sinh cosh exp log sqr

  SCOTTSZSB PARAMS=nnn    FN*FN FUNCTION=SIN/SIN BAILOUT=nnn+3

  SCOTTCZSB PARAMS=nnn    FN*FN FUNCTION=COS/COS BAILOUT=nnn+3

  SCOTTLTS PARAMS=nnn     FN*FN FUNCTION=LOG/SIN BAILOUT=nnn+3

  SCOTTLTC PARAMS=nnn     FN*FN FUNCTION=LOG/COS BAILOUT=nnn+3

                          Note: New functions for this type are
                          sin/cos sin/sinh sin/cosh sin/exp sin/sqr
                          cos/sinh cos/cosh cos/exp cos/sqr
                          sinh/sinh sinh/cosh sinh/exp sinh/log sinh/sqr
                          cosh/cosh cosh/exp cosh/log cosh/sqr
                          exp/exp exp/log exp/sqr log/log log/sqr sqr/sqr

  SCOTTSIC PARAMS=nnn     SQR(1/FN) FUNCTION=COS BAILOUT=nnn+3

  SCOTTSIS PARAMS=nnn     SQR(1/FN) FUNCTION=SIN BAILOUT=nnn+3

  TETRATE PARAMS=nnn      TETRATE BAILOUT=nnn+3

                          Note: New function type sqr(1/fn) with
                                sin cos sinh cosh exp log sqr

                          Note: New function type sqr(fn) with
                                sin cos sinh cosh exp log sqr
