Intrigued by Matchmaker, I sought during last winter to discover a mapping that allowed an arbitrary three quadratic Julia sets to be combined. One attempt was fairly successful and I called the result Triple Matchmaker.
It is defined by the iterated mapping
and the critical points are at -
a, 0, and
a, with another critical value at
d. Originally, I didn't have the final "+
d" on there, but at some point I decided to "Nova-ize" it to add to the variety of results it could produce.
I found it difficult to get arbitrary trios of quadratic Julias by random parameter values; I tended to get only zero-, one-, or two-attractor dynamics. So I turned to the massive, 8-dimensional parameter space to hunt, preferring to view the
a-plane while using random values for
b,
c, and
d. It's like a fantastical lava lamp in there, with multicolored blobs intersecting and occasionaly the distinctive bud-ribbon structures indicative of the potential for Herman rings.
Even then, finding three-attractor dynamics proved monstrously difficult, but I found plenty of gorgeous Julia sets with fewer. This one has only one attractor and looks like what I imagine you might see after binge drinking, or during a migraine, hence the name.
The hunt goes on, but the difficulty of dealing with the parameter space here means I shall be developing a whole new piece of software from the ground up to explore fractals. Development has begun as of June 26, 2009, with the evaluation at a REPL of
user=> (time (opt-mandelbrot-iteration -0.1 0.01 100000000))
"Elapsed time: 1412.76088 msecs"
:iters
which indicates that the core of a standard quadratic Mandelbrot iteration loop is 41% faster than the one in Ultra Fractal. (UF gets a pixel every two seconds, with periodicity off, zoomed into the heart of the Mandelbrot cardiod, per core, on the same machine.)
Even though it was running on a JVM rather than on the bare metal.
Impressive.
Now to build an actual image-generating program around it, and generalize it to more kinds of fractal. Eventually I'll be able to spawn automated searches for particular features, such as parameters with certain numbers of parameters, or implement an evolver, or ...
Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.
Detailed statistics:
Name: Bender
Date: February 25, 2009
Fractal: Triple Matchmaker Julia set
Location:
a = 2.7038648008 - 1.8796714841
i;
b = 0.1033577 + 0.0109489
i;
c = -1.42628 + 2.70015
i;
d = 0.
Depth: Very Shallow
Min Iterations: 87
Max Iterations: 674
Layers: 1
Anti-aliasing: 3x3, threshold 0.1, depth 1
Preparation time: 5 minutes
Calculation time: 2 minutes (2GHz dual-core Athlon XP)
Bender
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