Alternated hexagonal tiling honeycomb explained

bgcolor=#e7dcc3 colspan=2	Alternated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbols	h s 2s 2s s
Coxeter diagrams	↔ ↔ ↔ ↔
Cells
Faces
Vertex figure	truncated tetrahedron
Coxeter groups	{\overline{P}} _{3} , [3,3<sup>[3]] 1/2 {\overline{V}} _{3} , [6,3,3] 1/2 {\overline{Y}} _{3} , [3,6,3] 1/2 {\overline{Z}} _{3} , [6,3,6] 1/2 {\overline{VP}} _{3} , [6,3<sup>[3]] 1/2 {\overline{PP}} _{3} , [3<sup>[3,3]]
Properties	Vertex-transitive, edge-transitive, quasiregular

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: [6,3,3], [3,6,3], [6,3,6], [6,3<sup>[3]] and [3<sup>[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<sup>[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

bgcolor=#e7dcc3 colspan=2	Cantic hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h2
Coxeter diagrams	↔
Cells
Faces
Vertex figure	wedge
Coxeter groups	{\overline{P}} _{3} , [3,3<sup>[3]]
Properties	Vertex-transitive

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

bgcolor=#e7dcc3 colspan=2	Runcic hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h3
Coxeter diagrams	↔
Cells
Faces
Vertex figure	triangular cupola
Coxeter groups	{\overline{P}} _{3} , [3,3<sup>[3]]
Properties	Vertex-transitive

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

bgcolor=#e7dcc3 colspan=2	Runcicantic hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h2,3
Coxeter diagrams	↔
Cells
Faces
Vertex figure	rectangular pyramid
Coxeter groups	{\overline{P}} _{3} , [3,3<sup>[3]]
Properties	Vertex-transitive

The runcicantic hexagonal tiling honeycomb, h_2,3, or, has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs
• Semiregular honeycomb
• Hexagonal tiling honeycomb

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications,, (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapters 16–17: Geometries on Three-manifolds I,II)
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 https://link.springer.com/article/10.1007%2FBF01238563 https://web.archive.org/web/20140223225217/http://homeweb1.unifr.ch/kellerha/pub/TGarticle.pdf
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. http://www.sciencedirect.com/science/article/pii/S0024379501004773