# Order-5 hexagonal tiling honeycomb

Order-5 hexagonal tiling honeycomb

Perspective projection view
from center of Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbol {6,3,5}
Coxeter-Dynkin diagrams
Cells {6,3}
Faces hexagon {6}
Edge figure pentagon {5}
Vertex figure {3,5}
Dual Order-6 dodecahedral honeycomb
Coxeter group HV3, [6,3,5]
Properties Regular

In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is called paracompact because it has infinite cells. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is {3,5}, the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.[1]

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

A lower symmetry, [6,(3,5)*], index 120 construction exists with regular dodecahedral fundamental domains, and a icosahedral shaped Coxeter diagram with 6 axial infinite order (ultraparallel) branches.

## Images

It is similar to the 2D hyperbolic regular tiling, {∞,5}, with infinite apeirogonal faces, and 5 meeting around every vertex (peak).

## Related polytopes and honeycombs

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

There are 15 uniform honeycombs in the [5,3,6] Coxeter group family, including this regular form and its regular dual, order-5 hexagonal tiling honeycomb, {6,3,5}.

[6,3,5] family honeycombs
{6,3,5} r{6,3,5} t{6,3,5} rr{6,3,5} t0,3{6,3,5} tr{6,3,5} t0,1,3{6,3,5} t0,1,2,3{6,3,5}
{5,3,6} r{5,3,6} t{5,3,6} rr{5,3,6} 2t{5,3,6} tr{5,3,6} t0,1,3{5,3,6} t0,1,2,3{5,3,6}

It has a related alternation honeycomb, represented by , having icosahedron and triangular tiling cells.

It is a part of sequence of regular honeycombs with hexagonal tiling hyperbolic honeycombs of the form {6,3,p}:

This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:

### Rectified order-5 hexagonal tiling honeycomb

Rectified order-5 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{6,3,5} or t1{6,3,5}
2r{5,3[3]}
Coxeter diagrams
Cells {3,5}
r{6,3}, r{3[3]}
Faces Triangle {3}
Pentagon {5}
Hexagon {6}
Vertex figure
Pentagonal prism {}×{5}
Coxeter groups ${\displaystyle {\bar {VH}}_{3}}$, [6,3,5]
Properties Vertex-transitive, edge-transitive

The rectified order-5 hexagonal tiling honeycomb, t1{6,3,5}, has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.

It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.

r{p,3,5}
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name r{3,3,5}
r{4,3,5}

r{5,3,5}
r{6,3,5}

r{7,3,5}
... r{∞,3,5}

Image
Cells

{3,5}

r{3,3}

r{4,3}

r{5,3}

r{6,3}

r{7,3}

r{∞,3}

### Truncated order-5 hexagonal tiling honeycomb

Truncated order-5 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t{6,3,5} or t0,1{6,3,5}
Coxeter diagram
Cells {3,5}
t{6,3}
Faces Triangle {3}
Pentagon {5}
Hexagon {6}
Vertex figure
Pentagonal pyramid {}v{5}
Coxeter groups ${\displaystyle {\bar {VH}}_{3}}$, [6,3,5]
Properties Vertex-transitive

The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and triangular tiling facets, with a pentagonal pyramid vertex figure.

### Cantellated order-5 hexagonal tiling honeycomb

Cantellated order-5 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol rr{6,3,5} or t0,2{6,3,5}
Coxeter diagram
Cells r{3,5}
rr{6,3}
Faces Triangle {3}
Pentagon {5}
Hexagon {6}
Vertex figure
triangular prism
Coxeter groups ${\displaystyle {\bar {VH}}_{3}}$, [6,3,5]
Properties Vertex-transitive

The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron and rhombitrihexagonal tiling facets, with a triangular prism vertex figure.

### Bitruncated order-5 hexagonal tiling honeycomb

Bitruncated order-5 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol 2t{6,3,5} or t1,2{6,3,5}
Coxeter diagram
Cells
Faces
Vertex figure
Tetrahedron
Coxeter groups ${\displaystyle {\bar {VH}}_{3}}$, [6,3,5]
Properties Vertex-transitive

The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has a tetrahedral vertex figure.

### Cantitruncated order-5 hexagonal tiling honeycomb

Cantitruncated order-5 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol tr{6,3,5} or t0,1,2{6,3,5}
Coxeter diagram
Cells t{3,5}
tr{6,3}
Faces Pentagon {5}
Hexagon {6}
Dodecagon {12}
Vertex figure
triangular prism
Coxeter groups ${\displaystyle {\bar {VH}}_{3}}$, [6,3,5]
Properties Vertex-transitive

The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has truncated icosahedron and truncated trihexagonal tiling facets, with a tetrahedral vertex figure.

### Runcinated order-5 hexagonal tiling honeycomb

Runcinated order-5 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{6,3,5}
Coxeter diagram
Cells
Faces
Vertex figure
triangular antiprism
Coxeter groups ${\displaystyle {\bar {VH}}_{3}}$, [6,3,5]
Properties Vertex-transitive

The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has dodecahedron and truncated trihexagonal tiling facets, with a triangular antiprism vertex figure.

### Runcitruncated order-5 hexagonal tiling honeycomb

Runcitruncated order-5 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,3{6,3,5}
Coxeter diagram
Cells
Faces
Vertex figure
trapezoidal pyramid
Coxeter groups ${\displaystyle {\bar {VH}}_{3}}$, [6,3,5]
Properties Vertex-transitive

The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, has a trapezoidal pyramid vertex figure.

### Omnitruncated order-5 hexagonal tiling honeycomb

Omnitruncated order-5 hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{6,3,5}
Coxeter diagram
Cells
Faces
Vertex figure
tetrahedron
Coxeter groups ${\displaystyle {\bar {VH}}_{3}}$, [6,3,5]
Properties Vertex-transitive

The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, has a tetrahedral vertex figure.