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 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: [6,3,3], [3,6,3], [6,3,6], [6,3<sup>[3]</sup>] and [3<sup>[3,3]</sup>] , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)<sup>*</sup>] (remove 3 mirrors, index 24 subgroup); [3,6,3<sup>*</sup>] or [3<sup>*</sup>,6,3] (remove 2 mirrors, index 6 subgroup); [1<sup>+</sup>,6,3,6,1<sup>+</sup>] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3<sup>[3,3]</sup>]. The ringed Coxeter diagrams are , , , and , representing different types (colors) 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, .
The cantic hexagonal tiling honeycomb, h<sub>2</sub>{6,3,3}, or , is composed of octahedron, truncated tetrahedron, and trihexagonal tiling facets, with a wedge vertex figure.
The runcic hexagonal tiling honeycomb, h<sub>3</sub>{6,3,3}, or , has tetrahedron, triangular prism, cuboctahedron, and triangular tiling facets, with a triangular cupola vertex figure.
The runcicantic hexagonal tiling honeycomb, h<sub>2,3</sub>{6,3,3}, or , has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.