Cubic honeycomb

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

Cubic honeycomb
Type	Regular honeycomb
Family	Hypercube honeycomb
Indexing	J11,15, A1 W1, G22
Schläfli symbol	{4,3,4}
Coxeter diagram
Cell type	{4,3}
Face type	square {4}
Vertex figure	octahedron
Space group Fibrifold notation	Pm3m (221) 4−:2
Coxeter group	${\displaystyle {\tilde {C}}_{3}}$, [4,3,4]
Dual	self-dual Cell:
Properties	Vertex-transitive, regular

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.