Order-6 tetrahedral honeycomb

Order-6 tetrahedral honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{3,3,6} {3,3[3]}
Coxeter diagrams	↔
Cells	{3,3}
Faces	triangle {3}
Edge figure	hexagon {6}
Vertex figure	triangular tiling
Dual	Hexagonal tiling honeycomb
Coxeter groups	${\displaystyle {\overline {V}}_{3}}$, [3,3,6] ${\displaystyle {\overline {P}}_{3}}$, [3,3[3]]
Properties	Regular, quasiregular

In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling 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.