# 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 ${\displaystyle {\bar {P}}_{3}}$, [3,3[3]]
1/2 ${\displaystyle {\bar {V}}_{3}}$, [6,3,3]
1/2 ${\displaystyle {\bar {Y}}_{3}}$, [3,6,3]
1/2 ${\displaystyle {\bar {Z}}_{3}}$, [6,3,6]
1/2 ${\displaystyle {\bar {VP}}_{3}}$, [6,3[3]]
1/2 ${\displaystyle {\bar {PP}}_{3}}$, [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

It has 3 related form cantic hexagonal tiling honeycomb, , runcic hexagonal tiling honeycomb, , runcicantic hexagonal tiling honeycomb, .

### 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 ${\displaystyle {\bar {P}}_{3}}$, [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 ${\displaystyle {\bar {P}}_{3}}$, [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 h2,3{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 ${\displaystyle {\bar {P}}_{3}}$, [3,3[3]]
Properties Vertex-uniform

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