Honeycomb

A Tessellation in $n$-D, for $n\geq 3$. The only regular honeycomb in 3-D is $\{4, 3, 4\}$, which consists of eight cubes meeting at each Vertex. The only quasiregular honeycomb (with regular cells and semiregular Vertex Figures) has each Vertex surrounded by eight Tetrahedra and six Octahedra and is denoted $\left\{\matrix{\hfill 3\cr \hfill 3, 4}\right\}$.

There are many semiregular honeycombs, such as $\left\{\matrix{3, 3\hfill\cr 4\hfill}\right\}$, in which each Vertex consists of two Octahedra $\{3, 4\}$ and four Cuboctahedra $\left\{\matrix{3\cr 4\cr}\right\}$.

See also Sponge, Tessellation

References

Bulatov, V. ``Infinite Regular Polyhedra.'' http://www.physics.orst.edu/~bulatov/polyhedra/infinite/.