The Geometry Junkyard

Sierpinski Tetrahedra and Other Fractal Sponges

This seems to be everyone's favorite three-dimensional fractal, so much so that I've had to add a separate page for it and several other closely related fractals. The Sierpinski Tetrahedron has Hausdorff dimension two, so maybe it's not really a fractal in the "fractional dimension" sense of the word. It can be formed in many ways: (1) start with a single tetrahedron and remove octahedra from it, (2) recursively combine quadruples of tetrahedra into larger tetrahedra, (3) take "Pascal's Pyramid" of trinomial coefficients modulo two, (4) form the graph of the binary exclusive-or function on the unit square. The last construction shows that if you look down on it from the right direction, it just looks like a square, but from other viewpoints it has plenty of holes, so it can form a sort of "Venetian blind" that casts shadows only in certain directions.

From the Geometry Junkyard, computational and recreational geometry pointers.
Send email if you know of an appropriate page not listed here.
David Eppstein, Theory Group, ICS, UC Irvine.
Semi-automatically filtered from a common source file.