Alternated hexagonal tiling honeycomb


In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h, or, is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, as an alteration of a hexagonal tiling honeycomb.

Symmetry constructions

It has five alternated constructions from reflectional Coxeter groups all with four mirrors and only the first being regular: , , , ] and ], having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: ; or ; ; all of these are isomorphic to ]. The ringed Coxeter diagrams are,,, and, representing different types of hexagonal tilings in the Wythoff construction.

Related honeycombs

The alternated hexagonal tiling honeycomb has 3 related forms: the cantic hexagonal tiling honeycomb, ; the runcic hexagonal tiling honeycomb, ; and the runcicantic hexagonal tiling honeycomb,.

Cantic hexagonal tiling honeycomb

The cantic hexagonal tiling honeycomb, h2, or, is composed of octahedron, truncated tetrahedron, and trihexagonal tiling facets, with a wedge vertex figure.

Runcic hexagonal tiling honeycomb

The runcic hexagonal tiling honeycomb, h3, or, has tetrahedron, triangular prism, cuboctahedron, and triangular tiling facets, with a triangular cupola vertex figure.

Runcicantic hexagonal tiling honeycomb

The runcicantic hexagonal tiling honeycomb, h2,3, or, has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.