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.

**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.

© 1996-9

1999-05-26