Sponge

A sponge is a solid which can be parameterized by Integers $p$, $q$, and $n$ which satisfy the equation

$$2\sin\left({\pi\over p}\right)\sin\left({\pi\over q}\right)=\cos\left({\pi\over k}\right).$$

The possible sponges are $\{p, q \vert k\}=\{6, 6 \vert 3\}$, $\{6, 4 \vert 4\}$, $\{4, 6 \vert 4\}$, $\{3, 6 \vert 6\}$, and $\{4, 4 \vert \infty\}$ (Ball and Coxeter 1987).

See also Honeycomb, Menger Sponge, Sierpinski Sponge, Tetrix

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 152, 1987.

Cromwell, P. R. Polyhedra. New York: Cambridge University Press, p. 79, 1997.