Friday, June 1, 2007

Regular Sierpinski Polyhedra

I found an interesting article on the Sierpinski Polyhedra at "" entitled
"Regular Sierpinski Polyhedra1" by Aimee Kunnen and Steven Schlicker.

They made a detailed study on the Spierpinski polyhedra in general, and also report he fractal dimensions for five regular polyhedra:

1. Sierpinski tetrahedron: log(4)/log(2) = 2
2. Sierpinski hexahedron: log(8)/log(2) = 3
3. regular octahedron: log(6)/log(2) ≈ 2 585
4. Sierpinski dodecahedron: log(2)/log(d/d1) ≈ 2 32958
5. Sierpinski icosahedron: log(12)/log(d/d1) ≈ 2.581926
The meaning of d and d1 is basically the scaling factor I mentioned in the previous post, but for details, please check the original paper. The calculation of this ratio is a geometric problem, which can be solved in principle. Instead of doing complicated calculation, I just performed an estimate by straightforward inspection on the picture I obtained from the scanner.

Now the problem is what the value of log(d/d1) for the Spierpinski buckyball (Spierpinski truncated icosahedron) Chuang and his classmates made is. Or more generally, find out the contruction rules and the corresponding fractal structures, and dimension for Sierpinski Achimedean solids, etc.

No comments: