In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in an 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.
They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs.
It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, the octahedron comes from the rectified tetrahedron .