Order-6 cubic honeycomb

Order-6 cubic honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol	{4,3,6} {4,3[3]}
Coxeter diagram	↔ ↔
Cells	{4,3}
Faces	square {4}
Edge figure	hexagon {6}
Vertex figure	triangular tiling
Coxeter group	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {BP}}_{3}}$, [4,3[3]]
Dual	Order-4 hexagonal tiling honeycomb
Properties	Regular, quasiregular

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation (or honeycomb) in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

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.