quote:
Sierpinski Triangle ... Sierpinski Sponge
Just to nitpick, the technical terms are the Sierpinski Gasket, and the Menger Sponge. The Menger Sponge is the 3d equivalent of the Sierpinski Carpet.
One of the interesting thing about all of these shapes is that they can be modeled with iterative formulae in which, using any arbitrary starting point, it will converge on various positions in the gasket/carpet/sponge. Pick a random point (x,y for the gasket and carpet; x,y,z for the sponge). Plot it. Then, pick a random corner. Average your point with the corner point. Plot it. Repeat, each time choosing a random corner.
These shapes form the basis of more complex IFS (Iterative Function System) fractals; there are numerous IFS programs out there where you can change the parameters to make different IFS. It's kind of fun trying to get a particular shape (fern, maple leaf, your name, etc), especially the versions which allow you to add extra dimensions to it (color, frequency of impact), adjust the probability of choosing particular corners, scaling, and shear. Another neat thing of IFS is that you can create easy blend animations between different shapes.
I've been tempted recently to make myself a literal Sierpinski Carpet carpet, perhaps recursed to depth 3.
------------------
"Illuminant light,
illuminate me."
[This message has been edited by Rei, 10-20-2003]