Alternated hexagonal tiling honeycomb

Alternated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbols	h{6,3,3} s{3,6,3} 2s{6,3,6} 2s{6,3^[3]} s{3^[3,3]}
Coxeter diagrams	↔ ↔ ↔ ↔
Cells	tetrahedron Triangular tiling
Faces	Triangle {3}
Vertex figure	truncated tetrahedron
Coxeter groups	${\bar {P}}_{3}$ , [3,3^[3]] 1/2 ${\bar {V}}_{3}$ , [6,3,3] 1/2 ${\bar {Y}}_{3}$ , [3,6,3] 1/2 ${\bar {Z}}_{3}$ , [6,3,6] 1/2 ${\bar {VP}}_{3}$ , [6,3^[3]] 1/2 ${\bar {PP}}_{3}$ , [3^[3,3]]
Properties	Vertex-uniform, edge-transitive, quasiregular

In 3-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, or , with tetrahedron and triangular tiling cells, in an octahedron vertex figure. It is named by its construction as an alteration of a hexagonal tiling honeycomb.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

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

Cantic hexagonal tiling honeycomb

Cantic hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h₂{6,3,3}
Coxeter diagrams	↔
Cells	octahedron Truncated tetrahedron trihexagonal tiling
Faces	Triangle {3} Hexagon {6}
Vertex figure
Coxeter groups	${\bar {P}}_{3}$ , [3,3^[3]]
Properties	Vertex-uniform

The cantic hexagonal tiling honeycomb, h₂{6,3,3}, or .

Runcic hexagonal tiling honeycomb

Runcic hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h₃{6,3,3}
Coxeter diagrams	↔
Cells	cube triangular prism cuboctahedron Triangular tiling
Faces	Triangle {3} Hexagon {6}
Vertex figure
Coxeter groups	${\bar {P}}_{3}$ , [3,3^[3]]
Properties	Vertex-uniform

The runcic hexagonal tiling honeycomb, h₃{6,3,3}, or .

Runcicantic hexagonal tiling honeycomb

Runcicantic hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h_2,3{6,3,3}
Coxeter diagrams	↔
Cells	Truncated cube triangular prism Truncated octahedron trihexagonal tiling
Faces	Triangle {3} Square {4} Hexagon {6}
Vertex figure
Coxeter groups	${\bar {P}}_{3}$ , [3,3^[3]]
Properties	Vertex-uniform

The runcicantic hexagonal tiling honeycomb, h_2,3{6,3,3}, or .

References

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (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 [1] [2]
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [3]