© 2009 Mitch Richling
The Sierpinski gasket is a subset of the 2D Euclidean space that is the result of an infinite sequence of geometric operations with triangles. We begin the construction of the Sierpinski gasket with a single equilateral triangle:
The "center" of this triangle is removed. That is to say, an inverted, equilateral triangle of one half the height of the original triangle is removed from the center resulting in the following:
Next the centers of the remaining three triangles are removed, just as was done with the original triangle. This results in the following:
The result of continuing this process forever is the Sierpinski gasket. The resulting object can obviously never be physically drawn as we can't draw an infinite number of distinct lines! This is typical of a great many mathematical objects from simple ones like spheres to more complex ones like the Sierpinski gasket. These sort of constructions often lead to rather paradoxical properties for the objects in question. For example, the area of the Sierpinski gasket is zero while it perimeter is infinite.
This two dimensional process leading to the Sierpinski gasket can be generalized into the 3rd dimension. Start with a regular tetrahedron. Remove the maximal, inverted tetrahedron from the center. Remove the centers of the remaining tetrahedra in a similar way. Continue this process forever, and the result if the 3D Sierpinski gasket. The first few iterations of this process are illustrated here:
These objects are not easy to render. The result of iterating the process only 8 times yields an 82MB Povray input file containing more than half a million geometric primitives! Rendered with 5 light sources and with high quality settings, Povray takes well over an hour even on fast hardware circa 2010. Below are the results of using 5 iterations (both a still and a movie):
I strongly recommend the book "The Science of Fractal Images" to anyone who actually wants to use a computer to generate pictures. This book has an all star group of contributing authors.
If you are looking for a good introduction to dynamical systems, I would suggest Steven Storgatz's book: "Nonlinear Dynamics and Chaos" ISBN 0-7382-0453-6).
For a good, encyclopedic, introduction to the field in general, I strongly suggest "Chaos and Fractals: New Frontiers of Science" by Peitgen, Jurgens, and Saupe. This book is notable because of it's clear presentation and breadth of coverage. It's a great book to have around for the casual reader because it is broken up into semi self-contained sections that one can just pick up and read.
