Honeycomb

Remarks

Example: the cubes $[a, a+1] \times [b, b+1] \times [c, c+1]$ with $a, b, c \in \Z$ form the cubic honeycomb in $\R^3$. The cubes are convex polyhedra, every point $(x, y, z)$ lies in the cube with $(a, b, c) = (\floor{x}, \floor{y}, \floor{z})$, and distinct cubes have disjoint interiors.

Non-example: copies of a regular tetrahedron cannot form a honeycomb in $\R^3$. Take a generic point $p$ on an edge of one cell. The cells through $p$ each fill either a wedge with the dihedral angle $\alpha = \arccos(1/3)$ or a half-space, and these angles add up to $2\pi$. So $k \alpha = 2\pi$ or $k \alpha = \pi$ for some integer $k \ge 1$. Both fail, since $\alpha \approx 70.53^\circ$ and neither $360^\circ$ nor $180^\circ$ is an integer multiple of it.

Many authors also require the honeycomb to be face-to-face, so that two cells meet in a common face of both or not at all. The term is also used for tilings of hyperbolic space and of spheres by polytopes.