## Tetrix

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

 (1) (2) (3)

The Capacity Dimension is therefore
 (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.

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.