Written by Tom Brown in 1991. Uses a simple clipping algorithm to construct an animated image of each attractor. Great for a quick review of the attractors. There’s a large amount of documentation, written and video. Available from the Strange web site.
Now, the question is: what are these attractors? These are the results of chaos applied to a Cartesian plane. (And I get that this isn’t exactly a real space.)
In any event, the simplest answer is probably to go with ‚the most surprising results‘. Any graph with sufficiently high dimensionality (and in particular, high symmetry) will contain a whole sequence of infinite lines in it. And for such a graph, applying any transformation that ‚flows‘ along the lines will produce a sequence of points that ‚flows‘ along the lines.
For example, suppose you have a plane that’s symmetric about the y=x line. Then, applying the same transformation that you would apply to a chaotic map will produce a point that ‚flows‘ along this line. The example above would then flow along the lines $x=\sin y$, $y=\cos x$ – the triangle and cardioid.
Other examples would be a symmetric plane with an off-center point, or one with a parabola. Any odd-sided, convex polygon will have lines that ‚flow‘ in it. For example, consider a square, and take the line through one corner that goes to the opposite corner. Then apply the same transformation that you’d apply to an iterated map to the line segment connecting the two corners.
As a final example, the surface of a hyperbolic paraboloid.
So, even though they’re symmetric, these different attractors are not equivalent – they will produce different results when applied to the same initial condition.
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