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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /><title>Gnofract 4D</title><link rel="stylesheet" href="docbook.css" type="text/css" /><meta name="generator" content="DocBook XSL Stylesheets V1.75.2" /></head><body><div class="article" title="Gnofract 4D"><div class="titlepage"><div><div><h2 class="title"><a id="gnofract4d"></a>Gnofract 4D</h2></div><div><h3 class="subtitle"><i><span class="emphasis"><em>For when 2D fractals just aren't weird enough</em></span></i></h3></div><div><div class="author"><h3 class="author"><span class="firstname">Tim</span> <span class="surname">Whidbey</span></h3></div></div></div><hr /></div><div class="toc"><p><b>Table of Contents</b></p><dl><dt><span class="sect1"><a href="#introduction">Introduction</a></span></dt><dt><span class="sect1"><a href="#using">Using <span class="application">Gnofract 4D</span></a></span></dt><dd><dl><dt><span class="sect2"><a href="#basics">Interacting with the Fractal</a></span></dt><dt><span class="sect2"><a href="#files">Working with Files</a></span></dt><dt><span class="sect2"><a href="#tools">Tools</a></span></dt><dt><span class="sect2"><a href="#toolbar">Toolbar buttons</a></span></dt><dt><span class="sect2"><a href="#preferences">Changing Fractal Settings</a></span></dt><dt><span class="sect2"><a href="#hints">Hints</a></span></dt></dl></dd><dt><span class="sect1"><a href="#cmdref">Command Reference</a></span></dt><dd><dl><dt><span class="sect2"><a href="#MouseCommands">Mouse Commands</a></span></dt><dt><span class="sect2"><a href="#KeyboardShortcuts">Keyboard Shortcuts</a></span></dt></dl></dd><dt><span class="sect1"><a href="#maths"> About the maths</a></span></dt><dd><dl><dt><span class="sect2"><a href="#mset">The Mandelbrot Set</a></span></dt><dt><span class="sect2"><a href="#julia">The Julia Set</a></span></dt><dt><span class="sect2"><a href="#julibrot">The Julibrot</a></span></dt><dt><span class="sect2"><a href="#viewing">Viewing in Four Dimensions</a></span></dt><dt><span class="sect2"><a href="#hypercomplex">Hypercomplex Fractals and Quaternions</a></span></dt></dl></dd><dt><span class="sect1"><a href="#compiler">Writing Your Own Functions</a></span></dt><dd><dl><dt><span class="sect2"><a href="#frm_tutorial">Writing Your First Formula</a></span></dt></dl></dd><dt><span class="sect1"><a href="#formref">Formula Language Reference</a></span></dt><dd><dl><dt><span class="sect2"><a href="#Operators">Operators</a></span></dt><dt><span class="sect2"><a href="#Functions">Functions</a></span></dt><dt><span class="sect2"><a href="#Symbols">Symbols</a></span></dt></dl></dd><dt><span class="sect1"><a href="#internals"><span class="application">Gnofract 4D</span> Internals</a></span></dt><dd><dl><dt><span class="sect2"><a href="#layout">Source Code Layout</a></span></dt><dt><span class="sect2"><a href="#compiler_internals">Compiler</a></span></dt><dt><span class="sect2"><a href="#threading">Threading</a></span></dt></dl></dd><dt><span class="sect1"><a href="#bugs">Bugs and Known Issues</a></span></dt><dd><dl><dt><span class="sect2"><a href="#reporting">Reporting Bugs</a></span></dt><dt><span class="sect2"><a href="#compat">Backwards Compatibility</a></span></dt></dl></dd><dt><span class="sect1"><a href="#about">About <span class="application">Gnofract 4D</span></a></span></dt><dd><dl><dt><span class="sect2"><a href="#credits">Credits and copyright</a></span></dt></dl></dd></dl></div><div class="sect1" title="Introduction"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="introduction"></a>Introduction</h2></div></div></div><div class="epigraph"><p>
There is no excellent beauty which hath not some strangeness in the
proportion.</p><div class="attribution"><span>—<span class="attribution">Francis Bacon</span></span></div></div><p><span class="application">Gnofract 4D</span> is a program which draws complex mathematical
objects known as fractals, including the Mandelbrot and Julia sets and
many others. It allows you to treat a fractal which has more than one
parameter as a four-dimensional object and interactively view slices
of this object from arbitrary angles, giving rise to some very unusual
images.
</p><p>
This user's manual provides a tutorial introduction to <span class="application">Gnofract 4D</span> and the
mathematical background behind it, information on how to use the
graphical interface, and reference material on the language used to
write fractal formulas.
</p></div><div class="sect1" title="Using Gnofract 4D"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="using"></a>Using <span class="application">Gnofract 4D</span></h2></div></div></div><p>
<span class="application">Gnofract 4D</span>'s screen layout is deliberately simple. Most of the screen is
taken up by a viewport onto the fractal you're investigating. By
default, this is the Mandelbrot set. You can directly click on this to
zoom. The toolbar provides quick access to frequently used functions,
and more complex properties of the fractal are accessed through dialog
boxes brought up via the menu bar.
</p><p>
Initially, just play around - after all, generating fractals isn't
meant to be <span class="emphasis"><em>work</em></span>. If you make a change you don't
like, just hit Undo.
</p><div class="sect2" title="Interacting with the Fractal"><div class="titlepage"><div><div><h3 class="title"><a id="basics"></a>Interacting with the Fractal</h3></div></div></div><p>
Each fractal is an infinitely complex image, which you can see a slice
of in the main window. By left-clicking on the window, you can zoom in
to view finer and finer details. Just click on an area you like to
view it more closely. If you click and drag with the left button, you can
draw a white box around an area. When you let go, you zoom in
so that the area inside that box fills the window.
</p><p>
To zoom back out, click with the right button. You can also press
<span class="keycap"><strong>Home</strong></span> to return all parameters to the starting point
for this fractal, <span class="keycap"><strong>Control</strong></span>+<span class="keycap"><strong>Home</strong></span> to
reset the zoom only, or use <span class="guibutton">Undo</span> to go back one
step. There isn't a click and drag feature for the right button.
</p><p>
Clicking with the middle button rotates the view by 90 degrees in the
<span class="emphasis"><em>xz</em></span> and <span class="emphasis"><em>yw</em></span> axes. If you're
currently looking at the Mandelbrot set, you'll get a Julia set, and
vice versa. If you're looking at something else, you'll get something
weird. Note that clicking this twice doesn't take you back to where
you started: the screen also gets recentered on the point you clicked,
so clicking twice will normally give you a perturbed, deformed-looking
Mandelbrot.
</p><p>
The cursor keys pan around the image. Hold down
<span class="keycap"><strong>Control</strong></span>+ <span class="keycap"><strong><cursor></strong></span> to move more
quickly. Hold down <span class="keycap"><strong>Shift</strong></span> +
<span class="keycap"><strong><cursor></strong></span> to move around in the other two
dimensions, mutating the image. You can recenter the image on a point
by left-clicking on that point while holding down
<span class="keycap"><strong>Shift</strong></span>. </p><div class="note" title="Non-4D formulas" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Non-4D formulas</h3><p> Some fractal formulas
(typically those originally written for Fractint or UltraFractal)
don't support full 4D operation. (<span class="application">Gnofract 4D</span> determines this by whether
the formula uses the <code class="varname">#zwpixel</code> variable.) In this
case, the widgets for rotating in other dimensions, warping, and the
middle mouse button will be disabled. </p></div></div><div class="sect2" title="Working with Files"><div class="titlepage"><div><div><h3 class="title"><a id="files"></a>Working with Files</h3></div></div></div><p>
<span class="application">Gnofract 4D</span> uses several different types of file. These have different
purposes as listed in the table below.
</p><div class="informaltable"><table border="1"><colgroup><col /><col /><col /></colgroup><thead><tr><th>File Type</th><th>Extensions</th><th>Description</th></tr></thead><tbody><tr><td>
Parameter File
</td><td><code class="filename">.fct</code></td><td>
A parameter file is a small text file which contains all the settings
required to produce a particular image, such as the position of the
viewer along the X axis and the coloring scheme used. The parameter
file lists the formula used, but doesn't contain the entire formula,
so if you invent a new formula and want to share parameter files which
use it, you need to distribute the formula file as well.
<span class="application">Fractint</span> uses <code class="filename">.par</code> files for
this purpose and <span class="application">UltraFractal</span> uses <code class="filename">.upr</code>.
Unfortunately <span class="application">Gnofract 4D</span> can't read
those formats (yet).
</td></tr><tr><td>
Image File
</td><td>
<code class="filename">.jpg</code>,
<code class="filename">.png</code>
</td><td>
<span class="application">Gnofract 4D</span> supports JPEG and PNG file formats for image
output. <span class="emphasis"><em>No information about the fractal parameters is
stored in the image file</em></span>, so if you want to carry on
exploring from a particular point you need to save a parameter file as
well. <span class="application">Gnofract 4D</span> can't load image files, only save them. Choose
<span class="guimenu">File</span> > <span xmlns="" class="guimenuitem"><span xmlns="http://www.w3.org/1999/xhtml" class="guimenuitem">Save Image</span></span> to
save an image.
I recommend
using PNG images for high quality output, and JPEGs only when image
size is important, because JPEGs introduce artifacts which blur the
fine details of your fractal.
</td></tr><tr><td>
Formula File
</td><td>
<code class="filename">.frm</code>,
<code class="filename">.ufm</code>
</td><td>
A formula file is a collection of formulas, each of which is a
description of the algorithm used to draw a particular kind of
fractal, expressed in a simple programming language (see <a class="xref" href="#formref" title="Formula Language Reference">the section called “Formula Language Reference”</a> for language details). Both <span class="application">Gnofract 4D</span> and <span class="application">Fractint</span>
use <code class="filename">.frm</code> as the extension, and
<span class="application">UltraFractal</span> uses <code class="filename">.ufm</code>. In
general, any formula which works in <span class="application">Fractint</span> should work in <span class="application">Gnofract 4D</span>
and any which works in <span class="application">Gnofract 4D</span> should work in <span class="application">UltraFractal</span>, but the
reverse is not true.
</td></tr><tr><td>
Coloring Algorithm File
</td><td>
<code class="filename">.cfrm</code>,
<code class="filename">.ucl</code>
</td><td>
A coloring algorithm file is a collection of formulas used to assign
colors to a fractal. <span class="application">Gnofract 4D</span> combines a coloring algorithm with a
formula to produce the final image (this approach is shared with
<span class="application">UltraFractal</span> - <span class="application">Fractint</span> restricts you to built-in coloring
algorithms). Coloring algorithms are written in the same language as
fractal formulas. <span class="application">UltraFractal</span> uses the extension .ucl for its
coloring algorithm files. Some of these are compatible with <span class="application">Gnofract 4D</span> but
so far not very many.
</td></tr><tr><td>
Gradient File
</td><td>
<code class="filename">.map</code>,
<code class="filename">.ggr</code>
<code class="filename">.ugr</code>
</td><td>
A gradient file is a list of colors which is used to translate the
purely numerical output of the formula into something pretty to look
at. Gradients are currently saved only inside the fractal itself, not
as separate files. The GIMP uses the extension .ggr for its gradient
files; <span class="application">Fractint</span> uses .map for its own, simpler files. <span class="application">UltraFractal</span>
uses .ugr - these files contain multiple gradients.
</td></tr></tbody></table></div><p>
</p></div><div class="sect2" title="Tools"><div class="titlepage"><div><div><h3 class="title"><a id="tools"></a>Tools</h3></div></div></div><div class="sect3" title="Autozoom"><div class="titlepage"><div><div><h4 class="title"><a id="autozoom"></a>Autozoom</h4></div></div></div><p> Autozoom automatically searches for interesting parts of the
fractal by zooming in repeatedly, each time choosing the quadrant of
the screen which has the largest number of different colors (with some
randomization as well). You can start it going, go off for a coffee,
and see what it's found when you return, or guide it by clicking on
parts you like as it goes. It'll stop when the image reaches the
minimum size, which is set by default to stop just before you get to
the limits of the precision <span class="application">Gnofract 4D</span> offers.
</p></div><div class="sect3" title="Explorer"><div class="titlepage"><div><div><h4 class="title"><a id="explorer"></a>Explorer</h4></div></div></div><p>
The <span class="emphasis"><em>Explorer</em></span> helps you find neat-looking fractals
easily. It divides the screen into a large central section and smaller
"subfractals" which surround it. The central section is the main image
- you can click on this to zoom in, change the color, or perform any
operation you can normally. The other images around the edges are
"mutant" versions of the main image - they're formed by starting with
the base parameters and randomly changing them a bit. Whenever you
change the main image, you get a whole new set of mutants. If you like
a mutant more than the main picture, click on it to move it to the
middle - it then becomes the main picture and you get 12 new mutants
based on the new main image. When you're satisfied with the results,
click the Explorer button again to return to normal mode. </p><p> The Shape and Color sliders on the toolbar determines how
different the mutants are from the standard image. If Shape's set to
100, they're almost unrecognizable - if it's 0, they're exactly the
same. Similarly if Color's 100, each mutant is a different color, and
0 keeps the colors all the same. </p></div><div class="sect3" title="Formula Browser"><div class="titlepage"><div><div><h4 class="title"><a id="browser"></a>Formula Browser</h4></div></div></div><p>
The <span class="emphasis"><em>Formula Browser</em></span> allows you to look at all the
fractal formulas, coloring functions and gradients which are currently loaded
formula files. When you select a formula (from the Formula list in the
middle), the source window shows you the contents of that formula. You
can then use <span class="guibutton">Apply</span> to change the current
fractal to use that formula. This also resets the formula's parameters
to their defaults. Alternatively, <span class="guibutton">OK</span> applies
the formula and closes the window. </p><p>
Tips:
</p><div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem"><p> To load a new formula file, choose <span class="guimenu">File</span>
> <span xmlns="" class="guimenuitem"><span xmlns="http://www.w3.org/1999/xhtml" class="guimenuitem">Open Formula File</span></span>. </p></li><li class="listitem"><p> If you have changed a formula on disk, choose
<span class="guibutton">Refresh</span> to have <span class="application">Gnofract 4D</span> re-read
it. </p></li><li class="listitem"><p> If the formula contains errors,
<span class="guibutton">Apply</span> and <span class="guibutton">OK</span> will be
disabled. Check the Messages window to see what the errors
are. </p></li></ul></div><p>
</p></div><div class="sect3" title="Director"><div class="titlepage"><div><div><h4 class="title"><a id="director"></a>Director</h4></div></div></div><p>
The <span class="emphasis"><em>Director</em></span> allows you to create fractal movies.
You first define keyframes (including base keyframes) which are stops
in video. Then, for each of them, you define for how long they last
(<span class="guilabel">duration</span> - in frames), how long still image of
keyframe will stay in video (<span class="guilabel">stopped for</span>) and
interpolation type between several possibilities. When you hit
<span class="guibutton">Render</span> button, Director will render all frames
and put them in directory you selected and then it will try to create
video using <a class="ulink" href="http://www.transcoding.org/" target="_top"><span class="emphasis"><em>transcode</em></span></a>
tool. </p><p>
Tips:
</p><div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem"><p>
In order to end up with video file, not just a bunch of images, you need to have
<span class="emphasis"><em>transcode</em></span> tool compiled with support for ImageMagick.
</p></li><li class="listitem"><p>
You can always save your animation configuration for later use.
</p></li><li class="listitem"><p>
You can always stop rendering images. As long as you use same animation setting again
(for example, saving them before starting rendering), Director will starts from where
it stopped last time.
</p></li></ul></div><p>
</p></div><div class="sect3" title="Randomize Colors"><div class="titlepage"><div><div><h4 class="title"><a id="randomizecolors"></a>Randomize Colors</h4></div></div></div><p>
Replaces the current gradient with a randomly-generated new one.
</p></div><div class="sect3" title="Painter"><div class="titlepage"><div><div><h4 class="title"><a id="painter"></a>Painter</h4></div></div></div><p>
The painter dialog allows you to change the colors of your fractal by
clicking on the place where you want the color to be different. First,
select the color you want in the color selector. Then click on the
image - the part of the gradient most responsible for the color of
that pixel will be updated with the color you chose. Toggle the
"painting" button off if you want to interact with the fractal while
the painter dialog is up.
</p></div></div><div class="sect2" title="Toolbar buttons"><div class="titlepage"><div><div><h3 class="title"><a id="toolbar"></a>Toolbar buttons</h3></div></div></div><p>
On the left of the toolbar you can see a small preview window, which
updates as you change the angle or position buttons, to give you an
idea of what the fractal will look like when you release the button.
</p><p>
The first eight toolbar buttons correspond to the ten parameters which
define the view. The circular angle buttons, labelled
<span class="emphasis"><em>xy</em></span> to <span class="emphasis"><em>zw</em></span>, correspond to
rotation around the principal planes in four dimensions. They can
changed by dragging the dot around. When you let go, the fractal will
update. By the way, the <span class="emphasis"><em>zw</em></span> angle does work, you
just can't see its effects until you rotate in some other dimensions
first. </p><p>
The square position buttons, <span class="emphasis"><em>pan</em></span> and
<span class="emphasis"><em>wrp</em></span> (aka Warp), can be used to alter the view. The
<span class="emphasis"><em>pan</em></span> button allows you to pan around the current
view. The <span class="emphasis"><em>wrp</em></span> button allows you to move along the
other two axes, resulting in a mutated version of the current image.
Click inside one then drag the mouse, watching the preview window
update, then release the mouse when you like the results. </p><p>The warp menu allows even formulas which weren't designed to be
used with <span class="application">Gnofract 4D</span> to be used in 4D mode. If the current fractal has any
complex parameters, they're listed in this menu. If you select one,
that parameter's value is set to the value of the Z and W coordinates
for each pixel. Basically what this means is that the parameter you choose
becomes the fourth dimension. NB: If you set an explicit value for the parameter as
well, it'll be ignored.
</p><p>
The <span class="emphasis"><em>Deepen</em></span> button allows you to increase the current iteration count
and tighten the periodicity checking, for those occasions when the
auto-deepening and/or auto-tolerance doesn't get it right. This will
generally convert some 'inside' pixels to outside and make the image
look better, at the cost of longer rendering time. The image size list should be
self-explanatory. If you want a size not listed here, use the
Preferences dialog. </p><p> The <span class="emphasis"><em>Undo</em></span> and
<span class="emphasis"><em>Redo</em></span> buttons should be fairly obvious. You can
undo as many times as you like. Note that undo also affects parameters
such as color, not just position on screen. Lastly, the
<span class="emphasis"><em>Explore</em></span> button toggles Explorer Mode. See <a class="xref" href="#explorer" title="Explorer">the section called “Explorer”</a>.</p></div><div class="sect2" title="Changing Fractal Settings"><div class="titlepage"><div><div><h3 class="title"><a id="preferences"></a>Changing Fractal Settings</h3></div></div></div><p> In <span class="application">Gnofract 4D</span>, settings are divided into <span class="emphasis"><em>Fractal
Settings</em></span>, <span class="emphasis"><em>Gradients</em></span> and
<span class="emphasis"><em>Preferences</em></span>. <span class="emphasis"><em>Fractal
Settings</em></span> and <span class="emphasis"><em>Gradients</em></span> are saved in the
fractal's .fct file - they are properties of the fractal itself. By
contrast, <span class="emphasis"><em> Preferences</em></span> are your preferences for
<span class="application">Gnofract 4D</span>'s general behavior and are saved in <span class="application">Gnofract 4D</span>'s config file
(~/.gnofract4d), so they will still be active next time you
start <span class="application">Gnofract 4D</span> </p><div class="sect3" title="Fractal Settings"><div class="titlepage"><div><div><h4 class="title"><a id="settingsdialog"></a>Fractal Settings</h4></div></div></div><p>
The <span class="emphasis"><em>Formula</em></span> section allows you to choose the
formula used to calculate the fractal, and to set any parameters the
formula has. You can modify the formula by choosing <span class="guibutton">Browse
</span>, which invokes the Formula Browser. <span class="emphasis"><em>Max
Iterations</em></span> sets the number of iterations a point will go
through before we give up and assume it's a member of the
Julibrot. The other parameters on this pane are different depending on
the fractal type.
</p><p>
The <span class="emphasis"><em>Outer</em></span> page controls the function used to
decide what color to draw those points which aren't part of the
fractal set proper. Similarly, the <span class="emphasis"><em>Inner</em></span> page
controls the function used for points which are part of the set.
</p><p>
The <span class="emphasis"><em>Location</em></span> entryboxes allow you to
change the coordinates of the screen center and the image size.
The <span class="emphasis"><em>Angles</em></span> entryboxes allows you to set the rotation
angles. Only values between 0 and 2 * pi are different; values outside
this range "wrap" to points inside that range.
</p><p>
The <span class="emphasis"><em>Transforms</em></span> page allows you to control a list
of transformations applied to the image, and any parameters those transforms have.
</p><p>
The <span class="emphasis"><em>General</em></span> page gives a few options which don't
fit anywhere else. <span class="emphasis"><em>Flip Y Axis</em></span> causes Y to
increase top-to-bottom, rather than
bottom-to-top. <span class="emphasis"><em>Periodicity Checking</em></span> is a method
to speed up generation of the fractal. Points inside the fractal
eventually settle into a loop, where they repeatedly jump around
between the same points (they become 'periodic'). By noticing this, we
can skip calculating the point any further. You will generally want to
disable this if you are coloring the inside of the fractal, since it
will look rather weird otherwise. <span class="emphasis"><em>Tolerance</em></span> is
the distance between points which we'll accept as being 'the same' for
the purposes of periodicity detection. This is automatically adjusted
if the 'auto tolerance' setting in the preferences is enabled.
</p><p>
The <span class="emphasis"><em>Colors</em></span> tab allows you to edit the list of
colors used to display your fractal. For more complex gradient
editing, you can also use the GIMP's gradient editor.
</p></div><div class="sect3" title="Preferences"><div class="titlepage"><div><div><h4 class="title"><a id="prefsdialog"></a>Preferences</h4></div></div></div><div class="sect4" title="Image"><div class="titlepage"><div><div><h5 class="title"><a id="prefs_image"></a>Image</h5></div></div></div><p> <span class="emphasis"><em>Width</em></span> and <span class="emphasis"><em>Height</em></span> set
the size of the image in pixels. If <span class="emphasis"><em>Maintain Aspect
Ratio</em></span> is checked when you change either the width or
height, the other automatically changes to keep the image the same
shape. If <span class="emphasis"><em>Auto Deepen</em></span> is enabled, <span class="application">Gnofract 4D</span> will try
to automatically guess how many iterations are required to display the
image correctly. Similarly, <span class="emphasis"><em>Auto Tolerance</em></span>
adjusts the periodicity tolerance setting to try and calculate the
image quickly but correctly. <span class="emphasis"><em>Antialiasing</em></span> makes
the image look smoother but takes extra time to do. The difference
between 'fast' and 'best' is that fast antialiasing doesn't bother to
recalculate points which are the same color as their neighbors. This
speeds things up a lot but can miss a few details sometimes. </p></div><div class="sect4" title="Compiler"><div class="titlepage"><div><div><h5 class="title"><a id="prefs_compiler"></a>Compiler</h5></div></div></div><p>
<span class="application">Gnofract 4D</span> needs a C compiler to be available at runtime in order to work
(it dynamically creates the code to compute a particular formula when
you select it). The <span class="emphasis"><em>Compiler</em></span> page allows you to
specify a location for the compiler and options to pass to
it. <span class="emphasis"><em>If <span class="application">Gnofract 4D</span> is working fine, generally I suggest you leave
those settings alone. However you <span class="emphasis"><em>may</em></span> be able to
get noticeable performance gains by specifying the specific kind of
processor you have. For example, fairly modern AMD processors will
benefit by adding "-mathlon -msse2 -m3dnow" to the compiler flags.
</em></span> The <span class="emphasis"><em>Formula Search
Path</em></span> lists the directories where <span class="application">Gnofract 4D</span> will look for
formulas when a parameter file is loaded.
</p></div><div class="sect4" title="General"><div class="titlepage"><div><div><h5 class="title"><a id="prefs_general"></a>General</h5></div></div></div><p>
<span class="emphasis"><em>Number of threads</em></span> sets how many calculation
threads to use. Generally, leave this at 1 unless you have a
hyper-threaded or multi-processor computer, in which case set it to 1
greater than the number of CPUs you have.
</p></div><div class="sect4" title="Helpers"><div class="titlepage"><div><div><h5 class="title"><a id="prefs_helpers"></a>Helpers</h5></div></div></div><p>
<span class="application">Gnofract 4D</span> sometimes need to invoke a helper program. It'll try to use
your preferences from GConf if possible, and these fields are set to
be those defaults. However if that default is wrong somehow, you can
explicitly set it here.
</p></div><div class="sect4" title="Flickr"><div class="titlepage"><div><div><h5 class="title"><a id="prefs_flickr"></a>Flickr</h5></div></div></div><p>
<span class="application">Gnofract 4D</span> can upload your images to the online image-sharing service
Flickr. This page handles your online identity.
</p></div></div></div><div class="sect2" title="Hints"><div class="titlepage"><div><div><h3 class="title"><a id="hints"></a>Hints</h3></div></div></div><div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem"><p>If you zoom into a busy part of the fractal the image can look
"noisy". You can fix this by making the colors change more slowly - go
to the "Outer" tab and change the transfer function to 'sqrt' or 'log'
- or change "Density" to a number between 0 and 1 - a density of 0.1
makes the colors change 10 times more slowly.
</p></li><li class="listitem"><p>If you have an Inner coloring method other than zero, you may
see weird effects unless you disable periodicity checking.
</p></li></ul></div></div></div><div class="sect1" title="Command Reference"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="cmdref"></a>Command Reference</h2></div></div></div><div class="sect2" title="Mouse Commands"><div class="titlepage"><div><div><h3 class="title"><a id="MouseCommands"></a>Mouse Commands</h3></div></div></div><div class="informaltable"><table border="1"><colgroup><col /><col /></colgroup><thead><tr><th>Button</th><th>Action</th></tr></thead><tbody><tr><td>Left-click</td><td>Zoom in</td></tr><tr><td>Left-click and drag</td><td>Draw rectangle to zoom into.</td></tr><tr><td>Shift-Left-click</td><td>Recenter image on point clicked.</td></tr><tr><td>Middle-click</td><td>Flip to Julia set (or back to Mandelbrot).</td></tr><tr><td>Right-click</td><td>Zoom out.</td></tr><tr><td>Control-Right-click</td><td>Zoom out more quickly.</td></tr></tbody></table></div><p>
</p></div><div class="sect2" title="Keyboard Shortcuts"><div class="titlepage"><div><div><h3 class="title"><a id="KeyboardShortcuts"></a>Keyboard Shortcuts</h3></div></div></div><div class="informaltable"><table border="1"><colgroup><col /><col /></colgroup><thead><tr><th>Key</th><th>Action</th></tr></thead><tbody><tr><td>(arrow)</td><td>Pan image in indicated direction.</td></tr><tr><td>Ctrl+(arrow)</td><td>Pan more quickly in indicated direction.</td></tr><tr><td>Shift+(arrow)</td><td>Mutate image in Z or W directions.</td></tr><tr><td>Shift+Ctrl+(arrow)</td><td>Mutate more quickly.</td></tr><tr><td>Ctrl+1</td><td>Reset rotation to show the XY (Mandelbrot) plane.</td></tr><tr><td>Ctrl+2</td><td>Reset rotation to show the ZW (Julia) plane.</td></tr><tr><td>Ctrl+3</td><td>Reset rotation to show the XZ (Oblate) plane.</td></tr><tr><td>Ctrl+4</td><td>Reset rotation to show the XW (Parabolic) plane.</td></tr><tr><td>Ctrl+5</td><td>Reset rotation to show the YZ (Elliptic) plane.</td></tr><tr><td>Ctrl+6</td><td>Reset rotation to show the WY (Rectangular) plane.</td></tr><tr><td>Ctrl+A</td><td>Display AutoZoom dialog.</td></tr><tr><td>Ctrl+B</td><td>Display formula browser.</td></tr><tr><td>Ctrl+D</td><td>Display the Director (animation) window.</td></tr><tr><td>Ctrl+E</td><td>Enter (or leave) Explorer mode.</td></tr><tr><td>Escape</td><td>Quit full-screen mode.</td></tr><tr><td>Ctrl+F</td><td>Show fractal settings controls.</td></tr><tr><td>Home</td><td>Reset all numeric parameters to their defaults.</td></tr><tr><td>Ctrl+Home</td><td>Reset zoom to default level.</td></tr><tr><td>Ctrl+I</td><td>Save the current image to a file.</td></tr><tr><td>Ctrl+Shift+I</td><td>Add the current fractal to the render queue.</td></tr><tr><td>Ctrl+M</td><td>Launch an email editor with current image attached.</td></tr><tr><td>Ctrl+R</td><td>Create a new random color scheme.</td></tr><tr><td>Ctrl+Shift+S</td><td>Save the current parameters into a new file.</td></tr><tr><td>Ctrl+U</td><td>Upload the current image to Flickr.com.</td></tr><tr><td>Ctrl+Z</td><td>Undo the last operation.</td></tr><tr><td>Ctrl+Shift+Z</td><td>Redo an operation after undoing it.</td></tr><tr><td>F1</td><td>Show help file contents page.</td></tr><tr><td>F11</td><td>Show main window full-screen.</td></tr></tbody></table></div><p>
</p></div></div><div class="sect1" title="About the maths"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="maths"></a> About the maths</h2></div></div></div><div class="caution" title="Warning: Dubious mathematics ahead" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Warning: Dubious mathematics ahead</h3><p>I'm not a mathematician. You may find this discussion
insultingly oversimplified or just plain wrong.
</p></div><div class="sect2" title="The Mandelbrot Set"><div class="titlepage"><div><div><h3 class="title"><a id="mset"></a>The Mandelbrot Set</h3></div></div></div><p> The Mandelbrot may be
defined as the set of all complex numbers which, when you
repeatedly square them and add them again, never become infinite. (The
official definition of the set is somewhat different: it is the set of
points in the complex plane whose corresponding Julia sets are
connected. These end up being the same thing.)
We can tell that a number will eventually reach infinity if it ever
gets outside a circle of radius 2 around the origin. Unfortunately, we
can't tell in general that a point will <span class="emphasis"><em>never</em></span>
become infinite, so we have to estimate by trying a large number of
times before giving up.
</p><p>
In <span class="application">Gnofract 4D</span>, the formula is:
</p><pre class="programlisting">
Mandelbrot1 {
init:
z = 0
loop:
z = z^2 + c
bailout:
|z| < 4.0
}
</pre><p>
(|z| means the square of the magnitude of z). We calculate the loop
function repeatedly until the bailout condition is false or we've
performed the maximum number of iterations. At that point, if we
"bailed out", we know we're outside the set: otherwise we're
(probably) inside.
</p><p>
We do this repeatedly for each position on the screen, setting
<span class="emphasis"><em>c</em></span> to a different value for each point. This gives
rise to the familiar Mandelbrot set:
</p><p>
</p><div align="center"><img src="figures/mandelbrot_standard.png" align="middle" /></div><p>
</p><p>
All the points inside the set are (as is traditional) coloured
black. The points outside the set are different colours depending on
how long it takes them to escape from the set. These colours aren't
very mathematically significant, but they look nice.
</p><p>
So what happens if <span class="emphasis"><em>z</em></span> is initially set to a
complex value other than zero? (Strictly speaking, you shouldn't do
this. Zero is important because it is the <span class="emphasis"><em>critical
value</em></span> of z^2+c - other values are not mathematically
meaningful. However, as with most fractal programs, <span class="application">Gnofract 4D</span> allows you
to draw anything which looks interesting, regardless of its
mathematical purity.)
Well, you get a rather odd-looking, deformed M-set. This initial
value, which we'll call <span class="emphasis"><em>z0</em></span>, is called the intial
perturbation, and sets which have a non-zero <span class="emphasis"><em>z0</em></span>
are known as <span class="emphasis"><em>perturbed</em></span> sets:
</p><p>
</p><div align="center"><img src="figures/mandelbrot_perturbed.png" align="middle" /></div><p>
</p></div><div class="sect2" title="The Julia Set"><div class="titlepage"><div><div><h3 class="title"><a id="julia"></a>The Julia Set</h3></div></div></div><p>
The Julia set is actually drawn by the same procedure as the
Mandelbrot set. But instead of changing the value of
<span class="emphasis"><em>c</em></span> for each pixel, we keep <span class="emphasis"><em>c</em></span>
constant and change <span class="emphasis"><em>z0</em></span>. There is a different
Julia set for each value of <span class="emphasis"><em>c</em></span>; here's the one for
<span class="emphasis"><em>c</em></span>=0.
</p><p>
</p><div align="center"><img src="figures/julia_standard.png" align="middle" /></div><p>
</p><p>
Boring, isn't it? That's because we're just squaring the value at each
iteration without adding anything to it. So any value which starts
with a magnitude less than 1 will shrink forever (and hence is a
member of the set). All other values will grow forever, and so we've
just discovered a rather inefficient way of drawing perfect circles.
If we use a different value of <span class="emphasis"><em>c</em></span> we get something more
interesting:
</p><p>
</p><div align="center"><img src="figures/julia_perturbed.png" align="middle" /></div><p>
</p></div><div class="sect2" title="The Julibrot"><div class="titlepage"><div><div><h3 class="title"><a id="julibrot"></a>The Julibrot</h3></div></div></div><p>
Here we come to the heart of the matter. I said above that both the
Julia and Mandelbrot sets are drawn with the <span class="emphasis"><em>same
function</em></span>.
</p><pre class="programlisting">
julibrot(z0,c) {
init:
z = z0
loop:
z = z^2 + c
bailout:
|z| < 4.0
}
</pre><p>
The Julibrot function has two complex parameters, or four real
ones. In <span class="application">Gnofract 4D</span> I refer to the real parameters as x, y, z, and w:
these are c.re , c.im, z0.re and z0.im respectively.
The only difference is which points we choose to draw. To draw the
Mandelbrot set, we keep <span class="emphasis"><em>z0</em></span> constant and change
<span class="emphasis"><em>c</em></span> with each pixel. To draw the Julia set, we keep
<span class="emphasis"><em>c</em></span> constant and change <span class="emphasis"><em>z0</em></span>. If
you squint with your brain a bit, you can imagine both sets as
orthogonal "slices" through the same four-dimensional object. In
<span class="application">Gnofract 4D</span> terms, the Mandelbrot set is the <span class="emphasis"><em>xy</em></span>
plane, and the Julia set is the <span class="emphasis"><em>zw</em></span> plane. We can
also look at other planes: here's an image of the
<span class="emphasis"><em>xw</em></span> plane:
</p><p>
</p><div align="center"><img src="figures/xw_plane.png" align="middle" /></div><p>
</p></div><div class="sect2" title="Viewing in Four Dimensions"><div class="titlepage"><div><div><h3 class="title"><a id="viewing"></a>Viewing in Four Dimensions</h3></div></div></div><p>
However, we can draw any 2D slice we like, not just those which are
parallel to the Julibrot axes. To do this we'll need to describe our
scene by four things. First, the (<span class="emphasis"><em>x,y,z,w</em></span>)
coordinates of the center of the screen. Second, a vector for the
x-axis of the screen. This tells us how to change the parameters to
the Julibrot function as we proceed across the screen. Third, a vector
for the y-axis. Fourth and finally, the size of the image. For the
Mandelbrot set, our "default" view, the screen is centered at
[0,0,0,0], the x-vector is [1,0,0,0] and the y-vector is
[0,1,0,0]. The initial size is 4, because the whole Mandelbrot set
fits inside the 2x2 square. We can zoom into the set by changing
<span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>y</em></span> and the zoom factor.
</p><p>
If we want to draw other slices, we need to rotate our view through
four dimensions. In 3D, we can rotate in 3 directions: around the
<span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, and
<span class="emphasis"><em>z</em></span> axes. In 4D, we rotate around a <span class="emphasis"><em>
plane </em></span> rather than a line, and we can rotate in 6
directions: around the <span class="emphasis"><em>xy, xz, xw, yz, yw</em></span> and
<span class="emphasis"><em>zw</em></span> planes. For example, if we rotate through 90
degrees in the xz and yw directions, our screen vectors become
[0,0,1,0] and [0,0,0,1]: in other words, the Julia set. If we rotate
only part of the way, we get a "hybrid" between the two sets, which
looks decidedly odd:
</p><p>
</p><div align="center"><img src="figures/hybrid.png" align="middle" /></div><p>
</p><p>
In fact, we can rotate to any angle in each of the planes,
creating a whole world of bizarre pictures.
</p></div><div class="sect2" title="Hypercomplex Fractals and Quaternions"><div class="titlepage"><div><div><h3 class="title"><a id="hypercomplex"></a>Hypercomplex Fractals and Quaternions</h3></div></div></div><p> There are other kinds of fractal which are commonly described
as "four-dimensional" - hypercomplex and quaternion-based
fractals. Hypercomplex numbers have four components (one real and
three imaginary) where complex numbers have two. Since the
hypercomplex mandelbrot has two hypercomplex parameters, in <span class="application">Gnofract 4D</span>
terms it's actually an eight-dimensional object. <span class="application">Gnofract 4D</span> allows you to
set four of these as part of the view - the other four have to be set
via parameters. <span class="application">Gnofract 4D</span> doesn't support quaternions at present.</p></div></div><div class="sect1" title="Writing Your Own Functions"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="compiler"></a>Writing Your Own Functions</h2></div></div></div><p>
When you get tired of the fractal functions which come with Gnofract
4D, you can write your own, or take advantage of thousands of formulas
written by other fractal enthusiasts. Gnofract4D can load most fractal
formula files written for <span class="application">Fractint</span> (and some written for
<span class="application">UltraFractal</span>). However the compiler is not 100%
backwards-compatible with <span class="application">Fractint</span>, so unfortunately some fractals
can't be loaded, or will display differently when they do. <span class="application">Gnofract 4D</span>
also supports many constructs <span class="application">Fractint</span> doesn't, so you need to take
extra care when writing formulas if you want them to work in <span class="application">Fractint</span>
too.
</p><p>
Here are links to some online resources for formula files:
</p><div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem"><p> <a class="ulink" href="http://home.att.net/~Paul.N.Lee/OrgForm.html" target="_top">ORGFORM.ZIP</a>
A collection of about 25,000 <span class="application">Fractint</span> formula files by many authors,
originally compiled by George C. Martin and currently maintained by
Paul N. Lee. Indispensable. </p></li><li class="listitem"><p> <a class="ulink" href="http://formulas.ultrafractal.com/" target="_top"><span class="application">UltraFractal</span> public formula
database</a> Many thousands of formulas by users of
<span class="application">UltraFractal</span>. Most of these will work with <span class="application">Gnofract 4D</span>. Let me know of
any issues, since I aim to improve compatibility further in future
releases. </p></li></ul></div><p>
</p><div class="sect2" title="Writing Your First Formula"><div class="titlepage"><div><div><h3 class="title"><a id="frm_tutorial"></a>Writing Your First Formula</h3></div></div></div><p>
This section steps you through the creation of a new fractal
formula. By the way, the formulas for each of these steps can also be
found in the file <code class="filename">formulas/tutorial.frm</code>.
</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
Create a new file called '<code class="filename">example.frm</code>' (the
extension is important - <span class="application">Gnofract 4D</span> uses this to decide whether the file
is a formula or a coloring function).
</p></li><li class="listitem"><p>
Enter the following in <code class="filename">example.frm</code>.
</p><pre class="programlisting"><span><span class="bold"><strong>MyFormula {</strong></span></span><span>
</span><span><span class="emphasis"><em>; First example formula - this produces a variant on the Mandelbrot set</em></span></span><span>
</span><span><span class="bold"><strong>init:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>z</span><span> </span><span>=</span><span> </span><span><code class="literal">0</code></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>c</span><span> </span><span>=</span><span> </span><span>#pixel</span><span>
</span><span><span class="bold"><strong>loop:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>z</span><span> </span><span>=</span><span> </span><span>z</span><span>*</span><span>z</span><span>*</span><span>c</span><span> </span><span>+</span><span> </span><span>c</span><span>*</span><span>c</span><span>
</span><span><span class="bold"><strong>bailout:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>|</span><span>z</span><span>|</span><span> </span><span><</span><span> </span><span><code class="literal">4.0</code></span><span>
</span><span>}</span><span>
</span><span>
</span></pre><p>
</p></li><li class="listitem"><p>
Start <span class="application">Gnofract 4D</span>, choose <span xmlns="" class="guimenuitem"><span xmlns="http://www.w3.org/1999/xhtml" class="guimenuitem">File | Open Formula
File</span></span>, and open example.frm. You should see MyFormula in
the list of formulas to choose from. Select it and click Apply. You
should see an image like this:
</p><p>
</p><div><img src="figures/tutorial001.png" /></div><p>
</p><p>
A few things to note about the formula. It's divided into named
sections, marked with a colon: "init", "loop". and "bailout". The
compiler uses these to supply some of the standard scaffolding for a
fractal function so you don't have to. The "loop" statement is the
heart of the formula - this is the statement which is run repeatedly
and which defines the shape of your fractal.
</p></li><li class="listitem"><p>
At this point, the widgets for rotating the image in 4D will be
disabled, because your formula doesn't use any of the 4D
options. Let's turn those on. Edit your formula so it reads:
</p><pre class="programlisting"><span><span class="bold"><strong>MyFormula {</strong></span></span><span>
</span><span><span class="emphasis"><em>; Second example - introduce 4D</em></span></span><span>
</span><span><span class="bold"><strong>init:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>z</span><span> </span><span>=</span><span> </span><span><span class="bold"><strong>#zwpixel</strong></span></span><span> </span><span><span class="emphasis"><em>; take initial value from 4D position</em></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>c</span><span> </span><span>=</span><span> </span><span>#pixel</span><span>
</span><span><span class="bold"><strong>loop:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>z</span><span> </span><span>=</span><span> </span><span>z</span><span>*</span><span>z</span><span>*</span><span>c</span><span> </span><span>+</span><span> </span><span>c</span><span>*</span><span>c</span><span>
</span><span><span class="bold"><strong>bailout:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>|</span><span>z</span><span>|</span><span> </span><span><</span><span> </span><span><code class="literal">4.0</code></span><span>
</span><span>}</span><span>
</span><span>
</span></pre><p>
</p><p>
Then hit <span class="guibutton">Refresh</span> on the Formula Browser window. You
should now find that all the options are enabled. This is because the image now depends on all 4 components of the 4D space, via #pixel and #zwpixel.
</p></li><li class="listitem"><p>
Next let's add some parameters to our function:
</p><pre class="programlisting"><span><span class="bold"><strong>MyFormula {</strong></span></span><span>
</span><span><span class="emphasis"><em>; Third example - add a parameter</em></span></span><span>
</span><span><span class="bold"><strong>init:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>z</span><span> </span><span>=</span><span> </span><span>#zwpixel</span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>c</span><span> </span><span>=</span><span> </span><span>#pixel</span><span>
</span><span><span class="bold"><strong>loop:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>z</span><span> </span><span>=</span><span> </span><span><span class="bold"><strong>@myfunc</strong></span></span><span>(</span><span>z</span><span>*</span><span>z</span><span>*</span><span>c</span><span>)</span><span> </span><span>+</span><span> </span><span><span class="bold"><strong>@factor</strong></span></span><span> </span><span>*</span><span> </span><span>z</span><span> </span><span>+</span><span> </span><span>c</span><span>*</span><span>c</span><span>
</span><span><span class="bold"><strong>bailout:</strong></span></span><span>
</span><span> </span><span> </span><span> </span><span> </span><span>|</span><span>z</span><span>|</span><span> </span><span><</span><span> </span><span><code class="literal">4</code></span><span>
</span><span><span class="bold"><strong>default:</strong></span></span><span>
</span><span>param</span><span> </span><span>factor</span><span>
</span><span> </span><span>default</span><span> </span><span>=</span><span> </span><span>(</span><span><code class="literal">1.0</code></span><span>,</span><span><code class="literal">0.5</code></span><span>)</span><span>
</span><span>endparam</span><span>
</span><span>}</span><span>
</span></pre><p>
</p><p>
Hit <span class="guibutton">Refresh</span> again, then <span xmlns="" class="guimenuitem"><span xmlns="http://www.w3.org/1999/xhtml" class="guimenuitem">Edit |
Fractal Settings</span></span> to show the formula settings. You
should two extra parameters in addition to the standard "Max
Iterations" option: <span class="emphasis"><em>myfunc</em></span>, with a drop-down list
of functions, and <span class="emphasis"><em>fac</em></span> (or Factor) with a
draggable 4-way widget and 2 edit boxes. If you set myfunc to
<span class="emphasis"><em>sqr</em></span> and set factor to (-1,0.5) you should see:
</p><p>
</p><div><img src="figures/tutorial002.png" /></div><p>
</p><p>
Parameters like this are a quick way to add more options to your
fractal. Listing them in the "default" section is optional but
provides a way to pre-populate your formula with values that work
well. If you leave the default out <span class="application">Gnofract 4D</span> will use "ident" for
functions and 0 for numeric ones.
</p></li></ol></div><p>
</p></div></div><div class="sect1" title="Formula Language Reference"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="formref"></a>Formula Language Reference</h2></div></div></div><div class="sect2" title="Operators"><div class="titlepage"><div><div><h3 class="title"><a id="Operators"></a>Operators</h3></div></div></div><div class="informaltable"><table border="1"><colgroup><col /><col /><col /><col /></colgroup><thead><tr><th>Name</th><th>Description</th><th>Argument Types</th><th>Return Type</th></tr></thead><tbody><tr><td rowspan="4" align="left" valign="top">
!=</td><td rowspan="4" align="left" valign="top">
Inequality operator. Compare two values and return true if
they are different.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Bool, Bool
</td><td align="left" valign="top">
Bool</td></tr><tr><td rowspan="2" align="left" valign="top">
%</td><td rowspan="2" align="left" valign="top">
Modulus operator. Computes the remainder when x is divided by y. Not to be confused with the complex modulus.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
&&</td><td align="left" valign="top">
Logical AND.
</td><td align="left" valign="top">
Bool, Bool
</td><td align="left" valign="top">
Bool</td></tr><tr><td rowspan="6" align="left" valign="top">
*</td><td rowspan="6" align="left" valign="top">
Multiplication operator.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper, Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
Hyper, Float
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
Color, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td rowspan="5" align="left" valign="top">
+</td><td rowspan="5" align="left" valign="top">
Adds two numbers together.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper, Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
Color, Color
</td><td align="left" valign="top">
Color</td></tr><tr><td rowspan="5" align="left" valign="top">
-</td><td rowspan="5" align="left" valign="top">
Subtracts two numbers
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper, Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
Color, Color
</td><td align="left" valign="top">
Color</td></tr><tr><td rowspan="5" align="left" valign="top">
/</td><td rowspan="5" align="left" valign="top">
Division operator
</td><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex, Float
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper, Float
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
Color, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td rowspan="3" align="left" valign="top">
<</td><td rowspan="3" align="left" valign="top">
Less-than operator. Compare two values and return true if the first is less than the second.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Bool</td></tr><tr><td rowspan="3" align="left" valign="top">
<=</td><td rowspan="3" align="left" valign="top">
Less-than-or-equal operator. Compare two values and return true if the first is less than or equal to the second.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Bool</td></tr><tr><td rowspan="4" align="left" valign="top">
==</td><td rowspan="4" align="left" valign="top">
Equality operator. Compare two values and return true if they are
the same.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Bool, Bool
</td><td align="left" valign="top">
Bool</td></tr><tr><td rowspan="3" align="left" valign="top">
></td><td rowspan="3" align="left" valign="top">
Greater-than operator. Compare two values and return true if the first is greater than the second.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Bool</td></tr><tr><td rowspan="3" align="left" valign="top">
>=</td><td rowspan="3" align="left" valign="top">
Greater-than-or-equal operator. Compare two values and return true if the first is greater than or equal to the second.
</td><td align="left" valign="top">
Int, Int
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Bool</td></tr><tr><td rowspan="3" align="left" valign="top">
^</td><td rowspan="3" align="left" valign="top">
Exponentiation operator. Computes x to the power y.
</td><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex, Float
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
not</td><td align="left" valign="top">
Logical NOT.
</td><td align="left" valign="top">
Bool
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
||</td><td align="left" valign="top">
Logical OR.
</td><td align="left" valign="top">
Bool, Bool
</td><td align="left" valign="top">
Bool</td></tr></tbody></table></div></div><div class="sect2" title="Functions"><div class="titlepage"><div><div><h3 class="title"><a id="Functions"></a>Functions</h3></div></div></div><div class="informaltable"><table border="1"><colgroup><col /><col /><col /><col /></colgroup><thead><tr><th>Name</th><th>Description</th><th>Argument Types</th><th>Return Type</th></tr></thead><tbody><tr><td align="left" valign="top">
#rand</td><td align="left" valign="top">
Each time this is accessed, it returns a new pseudo-random complex number. This is primarily for backwards compatibility with Fractint formulas - use the random() function in new formulas.
</td><td align="left" valign="top">
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
#random</td><td align="left" valign="top">
Each time this is accessed, it returns a new pseudo-random complex number. This is primarily for backwards compatibility with Fractint formulas - use the random() function in new formulas.
</td><td align="left" valign="top">
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
@fn1</td><td align="left" valign="top">
Predefined function parameter used by Fractint formulas
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
@fn2</td><td align="left" valign="top">
Predefined function parameter used by Fractint formulas
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
@fn3</td><td align="left" valign="top">
Predefined function parameter used by Fractint formulas
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
@fn4</td><td align="left" valign="top">
Predefined function parameter used by Fractint formulas
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td rowspan="3" align="left" valign="top">
abs</td><td rowspan="3" align="left" valign="top">
The absolute value of a number. abs(3) = abs(-3) = 3.
abs() of a complex number is a complex number consisting of
the absolute values of the real and imaginary parts, i.e.
abs(a,b) = (abs(a),abs(b)).
</td><td align="left" valign="top">
Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td rowspan="3" align="left" valign="top">
acos</td><td rowspan="3" align="left" valign="top">
Inverse cosine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
acosh</td><td rowspan="3" align="left" valign="top">
Inverse hyperbolic cosine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
alpha</td><td align="left" valign="top">
The alpha component of a color. Can be assigned to.
</td><td align="left" valign="top">
Color
</td><td align="left" valign="top">
Float</td></tr><tr><td rowspan="3" align="left" valign="top">
asin</td><td rowspan="3" align="left" valign="top">
Inverse sine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
asinh</td><td rowspan="3" align="left" valign="top">
Inverse hyperbolic sine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
atan</td><td rowspan="3" align="left" valign="top">
Inverse tangent function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
atan2</td><td align="left" valign="top">
The angle between this complex number and the real line,
aka the complex argument.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td rowspan="3" align="left" valign="top">
atanh</td><td rowspan="3" align="left" valign="top">
Inverse hyperbolic tangent function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
blend</td><td align="left" valign="top">
Blend two colors together in the ratio given by the 3rd parameter.
</td><td align="left" valign="top">
Color, Color, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
blue</td><td align="left" valign="top">
The blue component of a color. Can be assigned to.
</td><td align="left" valign="top">
Color
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
bool</td><td align="left" valign="top">
Construct a boolean. It's not really required (bool x = bool(true) is just the same as bool x = true) but is included for consistency.
</td><td align="left" valign="top">
Bool
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
cabs</td><td align="left" valign="top">
The complex modulus of a complex number z.
cabs(a,b) is equivalent to sqrt(a*a+b*b).
This is also the same as sqrt(|z|)
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td rowspan="2" align="left" valign="top">
ceil</td><td rowspan="2" align="left" valign="top">
Round up to the next highest number.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td rowspan="2" align="left" valign="top">
cmag</td><td rowspan="2" align="left" valign="top">
The squared modulus of a complex or hypercomplex number z.
cmag(a,b) is equivalent to a*a+b*b. This is the same as |z|.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
color</td><td align="left" valign="top">
Constructs a new color from floating point red, green, blue and alpha
components. Equivalent to rgba.
</td><td align="left" valign="top">
Float, Float, Float, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
complex</td><td align="left" valign="top">
Construct a complex number from two real parts.
complex(a,b) is equivalent to (a,b).
</td><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
compose</td><td align="left" valign="top">
Composite the second color on top of the first, with opacity given
by the 3rd parameter.
</td><td align="left" valign="top">
Color, Color, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td rowspan="2" align="left" valign="top">
conj</td><td rowspan="2" align="left" valign="top">
The complex conjugate. conj(a,b) is equivalent to (a,-b).
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
cos</td><td rowspan="3" align="left" valign="top">
trigonometric sine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
cosh</td><td rowspan="3" align="left" valign="top">
Hyperbolic cosine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="2" align="left" valign="top">
cosxx</td><td rowspan="2" align="left" valign="top">
Incorrect version of cosine function. Provided for backwards
compatibility with equivalent wrong function in Fractint.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
cotan</td><td rowspan="3" align="left" valign="top">
Trigonometric cotangent function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
cotanh</td><td rowspan="3" align="left" valign="top">
Hyperbolic cotangent function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
exp</td><td rowspan="3" align="left" valign="top">
exp(x) is equivalent to e^x
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="2" align="left" valign="top">
flip</td><td rowspan="2" align="left" valign="top">
Swap the real and imaginary parts of a complex number.
flip(a,b) = (b,a).
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
float</td><td align="left" valign="top">
Construct a floating-point number.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td rowspan="2" align="left" valign="top">
floor</td><td rowspan="2" align="left" valign="top">
Round down to the next lowest number.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
fn1</td><td align="left" valign="top">
Predefined function parameter used by Fractint formulas
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
fn2</td><td align="left" valign="top">
Predefined function parameter used by Fractint formulas
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
fn3</td><td align="left" valign="top">
Predefined function parameter used by Fractint formulas
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
fn4</td><td align="left" valign="top">
Predefined function parameter used by Fractint formulas
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
gradient</td><td align="left" valign="top">
Look up a color from the default gradient.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
green</td><td align="left" valign="top">
The green component of a color. Can be assigned to.
</td><td align="left" valign="top">
Color
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
hsl</td><td align="left" valign="top">
Create a color from hue, saturation and lightness components. The alpha channel is set to to 1.0 (=100%).
</td><td align="left" valign="top">
Float, Float, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
hsla</td><td align="left" valign="top">
Create a color from hue, saturation and lightness components and an alpha channel.
</td><td align="left" valign="top">
Float, Float, Float, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
hsv</td><td align="left" valign="top">
Create a color from hue, saturation and value components. HSV is a similar color model to HSL but has a different valid range for brightness.
</td><td align="left" valign="top">
Float, Float, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
hue</td><td align="left" valign="top">
The hue of a color.
</td><td align="left" valign="top">
Color
</td><td align="left" valign="top">
Float</td></tr><tr><td rowspan="2" align="left" valign="top">
hyper</td><td rowspan="2" align="left" valign="top">
Construct a hypercomplex number with a real and 3 imaginary parts.
Can be passed either 2 complex numbers or 4 floating-point numbers.
hyper(a,b,c,d) is equivalent to the shorthand (a,b,c,d).
</td><td align="left" valign="top">
Float, Float, Float, Float
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
Complex, Complex
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
hyper_j</td><td align="left" valign="top">
The 3rd component of a hypercomplex number. Can be assigned to.
hyper_j(a,b,c,d) = c.
</td><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
hyper_jk</td><td align="left" valign="top">
The 3rd and 4th parts of a hypercomplex number.
Can be assigned to. hyper_jk(a,b,c,d) = (c,d).
</td><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
hyper_k</td><td align="left" valign="top">
The 4th component of a hypercomplex number. Can be assigned to.
hyper_k(a,b,c,d) = d.
</td><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
hyper_ri</td><td align="left" valign="top">
The real and imaginary parts of a hypercomplex number.
Can be assigned to. hyper_ri(a,b,c,d) = (a,b).
</td><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Complex</td></tr><tr><td rowspan="5" align="left" valign="top">
ident</td><td rowspan="5" align="left" valign="top">
Do nothing. ident(x) is equivalent to x.
This function is useless in normal formulas but
comes in useful as a value for a function parameter
to a formula. For example, a general formula like z = @fn1(z*z)+c
can be set back to a plain Mandelbrot by setting fn1 to ident.
Note: ident() is compiled out so there's no speed penalty involved.
</td><td align="left" valign="top">
Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Bool
</td><td align="left" valign="top">
Bool</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="2" align="left" valign="top">
imag</td><td rowspan="2" align="left" valign="top">
Extract the imaginary part of a complex or hypercomplex number.
imag(a,b) = b.
imag() is unusual in that it can be assigned to: imag(z) = 7 changes
the imag part of z.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
imag2</td><td align="left" valign="top">
The square of the imaginary part of a complex number.
real2(a,b) = b*b.
While not a generally useful function, this is provided to ease porting
of files from older Gnofract 4D versions.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
int</td><td align="left" valign="top">
Construct an integer. To convert a float to an int, use floor, ceil, round or trunc instead.
</td><td align="left" valign="top">
Int
</td><td align="left" valign="top">
Int</td></tr><tr><td rowspan="3" align="left" valign="top">
log</td><td rowspan="3" align="left" valign="top">
The natural log.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
lum</td><td align="left" valign="top">
The luminance (or brightness) of a color.
</td><td align="left" valign="top">
Color
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
manhattan</td><td align="left" valign="top">
The Manhattan distance between the origin and complex number z.
manhattan(a,b) is equivalent to abs(a) + abs(b).
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
manhattanish</td><td align="left" valign="top">
A variant on Manhattan distance provided for backwards
compatibility. manhattanish(a,b) is equivalent to a+b.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
manhattanish2</td><td align="left" valign="top">
A variant on Manhattan distance provided for backwards
compatibility. manhattanish2(a,b) is equivalent to (a*a + b*b)^2.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
max</td><td align="left" valign="top">
Returns the larger of its two arguments.
</td><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
max2</td><td align="left" valign="top">
max2(a,b) returns the larger of a*a or b*b. Provided for
backwards compatibility.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
mergemultiply</td><td align="left" valign="top">
Multiplies colors together. Result is always darker than either input.
</td><td align="left" valign="top">
Color, Color
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
mergenormal</td><td align="left" valign="top">
Returns second color, ignoring first.
</td><td align="left" valign="top">
Color, Color
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
min</td><td align="left" valign="top">
Returns the smaller of its two arguments.
</td><td align="left" valign="top">
Float, Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
min2</td><td align="left" valign="top">
min2(a,b) returns the smaller of a*a or b*b. Provided for
backwards compatibility.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td rowspan="4" align="left" valign="top">
neg</td><td rowspan="4" align="left" valign="top">
No documentation yet.
</td><td align="left" valign="top">
Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
rand</td><td align="left" valign="top">
Each time this is accessed, it returns a new pseudo-random complex number. This is primarily for backwards compatibility with Fractint formulas - use the random() function in new formulas.
</td><td align="left" valign="top">
</td><td align="left" valign="top">
Complex</td></tr><tr><td rowspan="2" align="left" valign="top">
real</td><td rowspan="2" align="left" valign="top">
Extract the real part of a complex or hypercomplex number.
real(a,b) = a.
real() is unusual in that it can be assigned to: real(z) = 7 changes
the real part of z.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
real2</td><td align="left" valign="top">
The square of the real part of a complex number.
real2(a,b) = a*a.
While not a generally useful function, this is provided to ease porting
of files from older Gnofract 4D versions.
</td><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Float</td></tr><tr><td rowspan="3" align="left" valign="top">
recip</td><td rowspan="3" align="left" valign="top">
The reciprocal of a number. recip(x) is equivalent to 1/x.
Note that not all hypercomplex numbers have a proper reciprocal.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td align="left" valign="top">
red</td><td align="left" valign="top">
The red component of a color. Can be assigned to.
</td><td align="left" valign="top">
Color
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
rgb</td><td align="left" valign="top">
Create a color from three color components. The alpha channel is set to to 1.0 (=100%).
</td><td align="left" valign="top">
Float, Float, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td align="left" valign="top">
rgba</td><td align="left" valign="top">
Create a color from three color components and an alpha channel.
</td><td align="left" valign="top">
Float, Float, Float, Float
</td><td align="left" valign="top">
Color</td></tr><tr><td rowspan="2" align="left" valign="top">
round</td><td rowspan="2" align="left" valign="top">
Round to the nearest number (0.5 rounds up).
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
sat</td><td align="left" valign="top">
The saturation of a color.
</td><td align="left" valign="top">
Color
</td><td align="left" valign="top">
Float</td></tr><tr><td rowspan="3" align="left" valign="top">
sin</td><td rowspan="3" align="left" valign="top">
trigonometric sine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
sinh</td><td rowspan="3" align="left" valign="top">
Hyperbolic sine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="4" align="left" valign="top">
sqr</td><td rowspan="4" align="left" valign="top">
Square the argument. sqr(x) is equivalent to x*x or x^2.
</td><td align="left" valign="top">
Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
sqrt</td><td rowspan="3" align="left" valign="top">
The square root.
The square root of a negative float number is NaN
(ie it is NOT converted to complex). Thus sqrt((-3,0)) != sqrt(-3).
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
tan</td><td rowspan="3" align="left" valign="top">
trigonometric sine function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="3" align="left" valign="top">
tanh</td><td rowspan="3" align="left" valign="top">
Hyperbolic tangent function.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td align="left" valign="top">
Hyper
</td><td align="left" valign="top">
Hyper</td></tr><tr><td rowspan="2" align="left" valign="top">
trunc</td><td rowspan="2" align="left" valign="top">
Round towards zero.
</td><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr><tr><td rowspan="3" align="left" valign="top">
zero</td><td rowspan="3" align="left" valign="top">
Returns zero.
</td><td align="left" valign="top">
Int
</td><td align="left" valign="top">
Int</td></tr><tr><td align="left" valign="top">
Float
</td><td align="left" valign="top">
Float</td></tr><tr><td align="left" valign="top">
Complex
</td><td align="left" valign="top">
Complex</td></tr></tbody></table></div></div><div class="sect2" title="Symbols"><div class="titlepage"><div><div><h3 class="title"><a id="Symbols"></a>Symbols</h3></div></div></div><div class="informaltable"><table border="1"><colgroup><col /><col /><col /><col /></colgroup><thead><tr><th>Name</th><th>Description</th><th>Argument Types</th><th>Return Type</th></tr></thead><tbody><tr><td align="left" valign="top">
#center</td><td align="left" valign="top">
Where the center of the image is located on the complex plane
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#color</td><td align="left" valign="top">
Set this from a coloring function to directly set the color instead of using a gradient
</td><td>Color</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#fate</td><td align="left" valign="top">
The fate of a point can be used to distinguish between different basins of attraction or whatever you like. Set this to a number from 2 to 128 to indicate that a different 'fate' has befallen this point. 0 indicates the point has diverged, 1 that it has been trapped, >1 whatever you like. Can only be usefully updated in the #final section.
</td><td>Int</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#index</td><td align="left" valign="top">
The point in the gradient to use for the color of this point.
</td><td>Float</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#inside</td><td align="left" valign="top">
Set this in the final section of a formula to override whether a point is colored with the inside or outside coloring algorithm. This is mainly useful in conjuction with #fate.
</td><td>Bool</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#magn</td><td align="left" valign="top">
The magnification factor of the image. This is the number of times the image size has doubled, or ln(4.0/size)
</td><td>Float</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#maxit</td><td align="left" valign="top">
No documentation yet.
</td><td>Int</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#maxiter</td><td align="left" valign="top">
No documentation yet.
</td><td>Int</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#numiter</td><td align="left" valign="top">
The number of iterations performed.
</td><td>Int</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#pi</td><td align="left" valign="top">
The constant pi, 3.14159...
</td><td>Float</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#pixel</td><td align="left" valign="top">
The (X,Y) coordinates of the current point. When viewing the Mandelbrot set, this has a different value for each pixel. When viewing the Julia set, it remains constant for each pixel.
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#solid</td><td align="left" valign="top">
Set this to true in a coloring function to use the solid color rather than the color map.
</td><td>Bool</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#tolerance</td><td align="left" valign="top">
10% of the distance between adjacent pixels.
</td><td>Float</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#xypixel</td><td align="left" valign="top">
The (X,Y) coordinates of the current point. When viewing the Mandelbrot set, this has a different value for each pixel. When viewing the Julia set, it remains constant for each pixel.
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#z</td><td align="left" valign="top">
No documentation yet.
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
#zwpixel</td><td align="left" valign="top">
The (Z,W) coordinates of the current point. (See #pixel for the other two coordinates.) When viewing the Mandelbrot set, this remains constant for each pixel on the screen; when viewing the Julia set, it's different for each pixel. Initialize z to some function of this to take advantage of 4D drawing.
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
@p1</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
@p2</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
@p3</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
@p4</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
@p5</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
@p6</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
maxit</td><td align="left" valign="top">
No documentation yet.
</td><td>Int</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
maxiter</td><td align="left" valign="top">
No documentation yet.
</td><td>Int</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
p1</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
p2</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
p3</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
p4</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
p5</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
p6</td><td align="left" valign="top">
Predefined parameter used by Fractint formulas
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
pi</td><td align="left" valign="top">
The constant pi, 3.14159...
</td><td>Float</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
pixel</td><td align="left" valign="top">
The (X,Y) coordinates of the current point. When viewing the Mandelbrot set, this has a different value for each pixel. When viewing the Julia set, it remains constant for each pixel.
</td><td>Complex</td><td class="auto-generated"> </td></tr><tr><td align="left" valign="top">
z</td><td align="left" valign="top">
No documentation yet.
</td><td>Complex</td><td class="auto-generated"> </td></tr></tbody></table></div></div></div><div class="sect1" title="Gnofract 4D Internals"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="internals"></a><span class="application">Gnofract 4D</span> Internals</h2></div></div></div><p>
This section explains how <span class="application">Gnofract 4D</span> is structured. You don't need to know
any of this to use the program, but it may come in handy if you want
to change it or contribute to its development (which you're heartily
encouraged to do).
</p><p>
<span class="application">Gnofract 4D</span> is implemented primarily in Python, with some C++
extensions. Extensive use
is made of Python unittest framework to keep everything working - each
Python file <code class="filename">foo.py</code> is accompanied by
<code class="filename">test_foo.py</code>, which contains unit tests for that
file's features. 'test.py' for each folder runs all of the tests.
</p><div class="sect2" title="Source Code Layout"><div class="titlepage"><div><div><h3 class="title"><a id="layout"></a>Source Code Layout</h3></div></div></div><p>
The important directories in the source are:
</p><div class="informaltable"><table border="1"><colgroup><col /><col /></colgroup><thead><tr><th>Directory</th><th>Contents</th></tr></thead><tbody><tr><td><p><code class="filename">fract4d</code></p></td><td> <p>This contains all the non-GUI-related, relatively
platform-independent parts of the code. This is in case it ever needs
to be ported to another environment (eg run on a server without a GUI
as part of a cluster). Most of the files here are parts of the
compiler (see below). The main class which represents a fractal is in
<code class="filename">fractal.py</code>. This holds references to the compiled
code, the formula and colorfunc definitions, the parameters and the
colormap. It also handles loading and saving information from a
<code class="filename">.fct</code> file, and provides
wrappers for some of the gnarlier C++ extension functions. </p>
</td></tr><tr><td><p><code class="filename">fract4d/c</code></p></td><td> <p>This contains the C++ extension code which is compiled
to produce <code class="filename">fract4dc.so</code>. This is divided into a
set of classes which communicate primaily via interfaces. The main
responsibility of this code is to call the 'pointFunc' (the function
which calculates a single pixel) once for each point on the
image. This code also does the bulk of the '4D' manipulation -
<code class="filename">vectors.h</code> contains code for 4-vectors and 4x4
matrix math. This library also handles multi-threaded calculations,
parcelling out the work to multiple MTFractWorkers via the queue in
<code class="filename">threadpool.h</code></p> </td></tr><tr><td><p><code class="filename">fract4dgui</code></p></td><td> <p>This contains the python code which implements the
GUI. It uses PyGTK as the GUI toolkit. The earliest PyGTK we support
is 1.99, which has some annoying incompatibilities with newer PyGTK's
like 2.4. I need to work out whether to ditch the older library
altogether or try to come up with some wrappers to hide the
differences. Basically there's one class per dialog or custom control,
and a few other for utility purposes. The central class is
<code class="classname">gtkfractal</code>, which wraps a
<code class="classname">fractal</code> and displays the results of the
calculation in a window. </p> </td></tr><tr><td><p><code class="filename">fract4dgui/c</code></p></td><td> <p>This contains the C code which implements the
fract4dguic.so extension. This only has one minimal function, to
obtain gconf settings. </p> </td></tr></tbody></table></div><p>
</p></div><div class="sect2" title="Compiler"><div class="titlepage"><div><div><h3 class="title"><a id="compiler_internals"></a>Compiler</h3></div></div></div><p>The most complicated part of <span class="application">Gnofract 4D</span> is the compiler. This takes
as input an UltraFractal or Fractint formula file, and produces C
code. We then invoke a C compiler (eg gcc) to produce a shared library
containing code to generate the fractal which we dynamically load.
</p><p>
The UltraFractal manual is the best current description of the formula
file format, though there are some UltraFractal features which are not
yet supported. You can download it from <a class="ulink" href="http://www.ultrafractal.com/uf3-manual.zip" target="_top">here</a>.
</p><p>
The implementation is based on the outline in <em class="citetitle">Modern Compiler Implementation in ML: basic
techniques</em> (Appel 1997, Cambridge). It doesn't do any
optimization at this point, leaving that to the C compiler used as a
back-end. It would be worthwhile to do some simple optimization (eg
constant-folding, removing multiplication by 1.0) because the C
compiler refuses to do this to floating point numbers.</p><p>
Overall structure: The <a class="ulink" href="http://www.dabeaz.com/ply/ply.html" target="_top">PLY</a> package
is used to do lexing and SLR parsing - it's in
<code class="filename">lex.py</code> and
<code class="filename">yacc.py</code>. <code class="filename">fractlexer.py</code> and
<code class="filename">fractparser.py</code> are the lexer and parser
definitions, respectively. They produce as output an abstract syntax
tree (defined in the <code class="classname">Absyn</code> module). The
<code class="classname">Translate</code> module type-checks the code,
maintains the symbol table (<code class="filename">symbol.py</code>) and
converts it into an intermediate form (<code class="filename">ir.py</code>).
<code class="classname">Canon</code> performs several simplifying passes on
the IR to make it easier to deal with, then
<code class="classname">codegen</code> converts it into a linear sequence of
simple C instructions. <code class="filename">stdlib.py</code> contains the
'standard library' of mathematical functions, like cosh(z). It's at
this point that complex and hypercomplex variables are expanded out
into pairs of floating point numbers - the C code is oblivious to the
complex numbers. Finally we invoke the C compiler to convert to a
native code shared library.</p><p>
At runtime the different phases happen at different times. First, the
entire .frm file is lexed and parsed. Then when a particular formula
is selected, it's translated and syntax-checked. The actual code is
only generated just before the fractal is drawn. This phase is
repeated whenever the function parameters are changed (eg @fn1 is set
to 'cosh').
</p><p>
Probably the ugliest part of the code is the handling of
parameters. Numeric parameters like floats are passed in as an array,
and the C++ code and Python code need to collaborate to work out which
indices into this array correspond to which params- this is done by
sorting them into alphabetic order. In general this area is a bit of a
mess.</p></div><div class="sect2" title="Threading"><div class="titlepage"><div><div><h3 class="title"><a id="threading"></a>Threading</h3></div></div></div><p>
One of the weirder parts of the code is how we deal with
threading. Basically we want the calculation of the fractal to happen
on a different thread (or multiple threads for SMP) from the main UI,
so you can interrupt at any point. This is complicated by the fact
that Python only allows a single thread in the Global Interpreter
Lock, and that PyGTK is often compiled by Linux distribution vendors
without thread support, meaning this lock is not released when running
the GTK main loop.
</p><p>
The way out of this is that the additional threads live only in the
C++ code, where they are invisible to the Python code and GTK. When
<code class="function">pycalc</code> is called with async=True, it spawns a
thread to do the calculation, which may in turn spawn more workers if
we want multiple threads. These all write to the image buffer and
report back what they're doing by writing messages into a pipe. This
pipe is added to the list of things the GTK main loop monitors, so
whenever a new message appears we get a callback into the gtkfractal
code, interleaved with the normal GTK events. We can interrupt a
calculation in progress by setting a var which the calculation threads
check frequently - they then abandon their work and quit. </p><div class="warning" title="Warning" style="margin-left: 0.5in; margin-right: 0.5in;"><h3 class="title">Warning</h3><p> Multiple threads and C++ exceptions do not coexist
well, at least on some of the libstdc++'s that <span class="application">Gnofract 4D</span> runs with. So the
C++ code can't throw exceptions or very odd things including crashes
will happen. </p></div></div></div><div class="sect1" title="Bugs and Known Issues"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="bugs"></a>Bugs and Known Issues</h2></div></div></div><div class="sect2" title="Reporting Bugs"><div class="titlepage"><div><div><h3 class="title"><a id="reporting"></a>Reporting Bugs</h3></div></div></div><p>
<span class="emphasis"><em>Please</em></span> report any bugs you encounter, either by
mailing <a class="ulink" href="mailto:catenary@users.sourceforge.net" target="_top">
catenary@users.sourceforge.net</a> or by
using the <a class="ulink" href="http://sourceforge.net/tracker/?func=add&group_id=785&atid=100785" target="_top">
bug form</a> at SourceForge. I'll do my best to fix them. </p></div><div class="sect2" title="Backwards Compatibility"><div class="titlepage"><div><div><h3 class="title"><a id="compat"></a>Backwards Compatibility</h3></div></div></div><p>
Version 2.x contains major architectural changes from version 1.x, so
unfortunately not all images generated by earlier versions will look
exactly the same or even load correctly. If you have a favorite image
which doesn't work any more, mail me the .fct file and I'll try and
convert it. I hope to fix most of these issues in future versions.
</p><p>
Specific known issues:
</p><div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem"><p> Files generated by <span class="application">Gnofract 4D</span> versions older than
1.4 can't be loaded. </p></li><li class="listitem"><p> Only one colormap per file is supported (the outer
one). Inner colormaps are ignored. </p></li></ul></div><p>
</p></div></div><div class="sect1" title="About Gnofract 4D"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="about"></a>About <span class="application">Gnofract 4D</span></h2></div></div></div><p>
This is <span class="application">Gnofract 4D</span> version 3.14.1. You can find the most recent version of
<span class="application">Gnofract 4D</span> from <a class="ulink" href="http://gnofract4d.sourceforge.net" target="_top">
http://gnofract4d.sourceforge.net/</a>.
</p><div class="sect2" title="Credits and copyright"><div class="titlepage"><div><div><h3 class="title"><a id="credits"></a>Credits and copyright</h3></div></div></div><p>
<span class="application">Gnofract 4D</span> is Copyright 1999-2009 Tim Whidbey <a class="ulink" href="mailto:catenary@users.sourceforge.net" target="_top">(catenary@users.sourceforge.net)
</a>, and is distributed under the <span class="emphasis"><em>BSD
license</em></span>. See the file "COPYING" for details.
</p><p>
<span class="application">Gnofract 4D</span> was originally based on Gnofract, written by Aurelien Alleaume
<a class="ulink" href="mailto:manchot@club-internet.fr" target="_top">(manchot@club-internet.fr)
</a>,
though none of the original code remains in the current version.
Gnofract could once be obtained from
<a class="ulink" href="http://www.multimania.com/mason/" target="_top">
http://www.multimania.com/mason/</a> but this no longer appears to
work.
</p><p>
Branko Kokanovic developed and contributed the animation
feature. Chris Le Sueur provided parts of the gradient editing
feature. Henryk Trappmann provided HSV gradient support.
The man page was contributed by Aleksander Adamowski.</p><p>
The formula language which <span class="application">Gnofract 4D</span> uses originated in <span class="application">Fractint</span> and
was substantially enhanced in <span class="application">UltraFractal</span>. However the compiler
implementation does not share any code with those programs.
</p><p>
The <span class="application">Gnofract 4D</span> distribution contains palette (.map) files by a number of
authors which were originally distributed with <a class="ulink" href="http://spanky.triumf.ca/" target="_top"><span class="application">Fractint</span></a> under somewhat murky
licensing conditions. It also contains a copy of "standard.ucl",
originally distributed with <a class="ulink" href="http://www.ultrafractal.com/" target="_top"><span class="application">UltraFractal</span></a>, by kind
permission of Frederik Slijkerman, Damien Jones, and Kerry Mitchell.
"blatte1.ugr" and "blatte2.ugr" are included by kind permission of
<a class="ulink" href="http://exoteric.roach.org/" target="_top">'Blatte'</a>. The formulas
in Sterling2.frm are translations of formulas originally created by
Tad Boniecki for use with the SterlingWare 2 fractal program.
</p><p>
<code class="filename">gmpy.c</code> and <code class="filename">gmpy.h</code> are from
the GMPY package (http://gmpy.sf.net), and are distributed under the
LGPL license. </p><p>
<code class="filename">lex.py</code> and <code class="filename">yacc.py</code> come from
the PLY package, and are distributed under the LGPL license. </p><p>
Some of the menu icons are taken or adapted from
the <a class="ulink" href="http://tango.freedesktop.org" target="_top">Tango icon set</a>.
</p></div></div></div></body></html>
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