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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

$\displaystyle N_n$ $\textstyle =$ $\displaystyle 4^n$ (1)
$\displaystyle L_n$ $\textstyle =$ $\displaystyle ({\textstyle{1\over 2}})^n=2^{-n}$ (2)
$\displaystyle A_n$ $\textstyle =$ $\displaystyle {L_n}^3N_n = ({\textstyle{1\over 2}})^n.$ (3)

The Capacity Dimension is therefore
$\displaystyle d_{\rm cap}$ $\textstyle =$ $\displaystyle -\lim_{n\to \infty}{\ln N_n\over\ln L_n} = -\lim_{n\to\infty}{\ln(4^n)\over\ln(2^{-n})}$  
  $\textstyle =$ $\displaystyle {\ln 4\over\ln 2} = {2\ln 2\over\ln 2} = 2,$ (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.

\begin{figure}\begin{center}\BoxedEPSF{TetrixRotation.epsf scaled 1050}\end{center}\end{figure}

See also Menger Sponge, Sierpinski Sieve


Dickau, R. M. ``Sierpinski Tetrahedron.''

Eppstein, D. ``Sierpinski Tetrahedra and Other Fractal Sponges.''

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© 1996-9 Eric W. Weisstein