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 alternation of a hexagonal [tiling honeycomb].
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.
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.