Alternated hexagonal tiling honeycomb

In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, 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: [6,3,3],  [3,6,3],  [6,3,6],  [6,3[3]] and [3[3,3]], having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are, , , and , representing different types (colors) 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{6,3,3}, 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{6,3,3}, 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{6,3,3}, or, has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.