Tetrix

The 3-D analog of the Sierpinski Sieve illustrated above, also called the Sierpinski Sponge or Sierpinski Tetrahedron. Let $N_n$ be the number of tetrahedra, $L_n$ the length of a side, and $A_n$ the fractional Volume of tetrahedra after the $n$th iteration. Then

$$N_n = 4^n$$ (1)

$$L_n = ({\textstyle{1\over 2}})^n=2^{-n}$$ (2)

$$A_n = {L_n}^3N_n = ({\textstyle{1\over 2}})^n.$$ (3)

The Capacity Dimension is therefore

$$d_{\rm cap} = -\lim_{n\to \infty}{\ln N_n\over\ln L_n} = -\lim_{n\to\infty}{\ln(4^n)\over\ln(2^{-n})}$$

$$= {\ln 4\over\ln 2} = {2\ln 2\over\ln 2} = 2,$$ (4)

so the tetrix has an Integral Capacity Dimension (albeit one less than the Dimension of the 3-D Tetrahedra from which it is built), despite the fact that it is a Fractal.

The following illustration demonstrates how this counterintuitive fact can be true by showing three stages of the rotation of a tetrix, viewed along one of its edges. In the last frame, the tetrix ``looks'' like the 2-D Plane.

See also Menger Sponge, Sierpinski Sieve

References

Dickau, R. M. ``Sierpinski Tetrahedron.'' http://forum.swarthmore.edu/advanced/robertd/tetrahedron.html.

Eppstein, D. ``Sierpinski Tetrahedra and Other Fractal Sponges.'' http://www.ics.uci.edu/~eppstein/junkyard/sierpinski.html.