# Tetrahedral-icosahedral honeycomb

Tetrahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Semiregular honeycomb
Schläfli symbol {(3,3,5,3)}
Coxeter diagram or or
Cells {3,3}
{3,5}
r{3,3}
Faces triangular {3}
pentagon {5}
Vertex figure
icosidodecahedron
Coxeter group [(5,3,3,3)]
Properties Vertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in a icosidodecahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

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.

It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, rectified tetrahedral r{3,3}, becomes the regular octahedron {3,4}.

## Images

 Centered on octahedron