These Fractal demos compliment Eric Giguere's fractal article.  They can all
be run by double-clicking on their icons and can be exited by clicking the
close gadget in the upper left corner of the screen.

Chaos by Eric Giguere
    This program follows the algorithm described in the article as "the Chaos
Game", using random points to create a "Sierpinski Triangle".  The program
will run as long as you want, slowly filling in empty spaces in the
pattern.  Since the points are filled at random, the pattern fills
inefficiently; as more points are drawn, there is a greater likelyhood that
the next point will fall on a point which is already lit.  This program uses
random points following a simple rule to make a "self-similar" object,  a
fascinating contrast is the "TriTree" fractal drawn by the PolyFrac program.
This duplicates the Sierpinski triangle (more or less) using an explicitly
self-similar algorithm, and is much faster as a result.  Fractal ferns and
other replications of natural objects use a reverse of the PolyFrac technique
which is known as the "collage theorem": one draws a small pattern and tries
to fit different sizes of that pattern onto the object that is to be
reproduced.  When a pattern is found which matches the object pretty well on
several scales, it is used as the "generator" for the picture of the object.
When real ferns grow, they use a chaos game type method where random factors
influence the growth of each cell and leaf, but simple governing rules cause
the outcome to be a familiar fern pattern.

GBMan by Eric Giguere
 Gingerbread Man   Say you have two arrays of points, call them x and y.
 The first element in each array is set to some (real) pre-defined value
 between -1 and 1 and the remaining elements are defined by the equations;

         x[ n ] = 1 - y[ n-1 ] + | x[ n-1 ] |
         y[ n ] = x[ n-1 ]

 In this formula, x[ n-1 ] means the absolute (unsigned) value of the array
element x[ n-1 ]. Plot a point at each location ( x[ n ], y[ n ] ) on a piece
of graph paper.  This is what the GBMan program does. (The
default starting position is (-0.10051202, 0.00).  Alter the starting point
slightly and the picture changes, sometimes dramatically.
 This is known as the "butterfly effect", or sensitive dependence on initial
conditions. Small changes in initial conditions lead to large differences
later on. GBMan can be run from the CLI with optional arguments specifying
the starting point. Just type
                 GBMan X Y
where X and Y are the starting points.  I modified Eric's code to accept
different starting positions, expecting exciting new pictures, I was
disappointed.  I didn't experiment too much, so maybe there are exciting
other pictures to make...

Henon
 Not all equations create such arresting images. Consider the Henon Map,
given by the equations:
     x[ n ] = 1 + y[ n-1 ] - 1.4 * x[ n-1 ] * x[ n-1 ]
     y[ n ] = 0.3 * x[ n ]

with a default starting point of (-1.00, 0.00),
While not as impressive as the gingerbread man, the Henon Map demonstrates
another fractal phenomenon -- the strange attractor, a set of points which
seems to "attract" all other points. The strange attractor is a
well-documented, but not well-understood mathematical phenomenon.
Henon can also be run from the CLI with optional arguments specifying the
starting point.  Just type:
                   Henon X Y
where X and Y are the starting points.  This also proved rather disappointing
for the few values I tried, but I encourage everyone to try it out.  People
with C compilers can also change the other parameters to see what happens.


Julia by Eric Giguere

Make sure you have the file mathtrans.library in your LIBS: directory before
running Julia or either of the Mandelbrot programs.  The Julia (Julia-Fatou)
set is closely related to the Mandelbrot set; for every point in the
Mandelbrot set, there is an entire Julia set picture representing two other
variables.  CLI arguments can be used with this program.  The default
starting position is (-1.0 , 0.0).  The best results obtain when the X
coordinate is negative.

Mandelbrot and Mandelbrot32 by Eric Giguere

These two programs draw the Mandelbrot set.  The color at each point in the
plane is determined by the number of times the process
                            Z -> Z*Z + Z
can be repeated before Z ( which is X + i*Y ) goes above or below a certain
limit (which determines whether it is in the Mandelbrot set or not).  These
programs can take a long time to finish.  Mandelbrot32 is a version of
Mandelbrot which opens a 32-color screen rather than a 16-color interlaced
screen like Mandelbrot.

Dragon, Hilbert, PolyFrac  by Daniel Wolf Ph.D.

These three programs build self-similar fractal drawings.  The basic scenario
goes like this -- a straight line segment (the "initiator") is replaced by a
certain shape (the "generator"). This shape is actually the first picture
drawn by these programs.  The computer then replaces each line segment in the
generator with a smaller version of the entire generator.  This process is
repeated until the next level becomes too small to see. Dragon and Hilbert
are optimized, stand-alone versions of the dragon and Hilbert curves drawn by
PolyFrac. These three programs are capable of checking whether the Workbench
screen is interlaced, to determine how many levels of recursion to use.
PolyFrac is provided courtesy of COMPUTE! Books. Source code for PolyFrac can
be found in "COMPUTE!'s Amiga Machine Language Programming Guide" by Daniel
Wolf and Douglas Leavitt, copyright 1988 by COMPUTE! Books.

The source code for the above programs is in the FractalSource subdirectory
of the Programming drawer.

StrangeTraj, AttractorTime  by Arnie Cachelin (me!)

These two programs model and display the dynamics of a system of non-linear
differential equations which exhibit chaotic behavior.  This behavior can be
viewed in a graph showing the system's "phase-space trajectory."  This
trajectory shows an "attractor" where the system seems to spiral in toward a
final position.  Since the system erratically jumbs between attractors,
without ever stopping in the center, it is called a "strange attractor".
This system has three variables or dimensions in phase-space: X, Y, and Z.
Chaotic systems demonstrate "sensitive dependence on initial conditions"
which means that two identical systems with slightly different starting
positions will soon have totally different trajectories. These programs can
be run from the Workbench or from CLI with optional arguments to specify the
initial position (in 3-D) and the number of time steps to go through, for
example type:
          AttractorTime 1 1 .5 4000

to start the evolution at the point (1,1,0.5) and step through 4000 time
steps. StrangeTraj, shows the trajectories characteristic of a strange
attractor in three windows which give three different perspectives. The other
program, AttractorTime,  shows the X, Y and Z positions versus time.  Notice
the regular oscillations punctuated with irregular jumps.  Try varying the
starting conditions by a little bit and see how soon the trajectories differ.
Although the tangled spirals will look the same after a long time, the order
in which they are filled is different and pictures drawn with only a few
hundred steps should differ considerably.