Square tiling honeycomb

Square tiling honeycomb
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{4,4,3} r{4,4,4} {41,1,1}
Coxeter diagrams	↔ ↔ ↔
Cells	{4,4}
Faces	square {4}
Edge figure	triangle {3}
Vertex figure	cube, {4,3}
Dual	Order-4 octahedral honeycomb
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3] ${\displaystyle {\overline {N}}_{3}}$, [43] ${\displaystyle {\overline {M}}_{3}}$, [41,1,1]
Properties	Regular

In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.^[1]

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.