# Order-5 cubic honeycomb

Order-5 cubic honeycomb

Poincaré disk models
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {4,3,5}
Coxeter diagram
Cells {4,3}
Faces square {4}
Edge figure pentagon {5}
Vertex figure
icosahedron
Coxeter group ${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
Dual Order-4 dodecahedral honeycomb
Properties Regular

The order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral 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.

## Description

It is analogous to the 2D hyperbolic order-5 square tiling, {4,5}
 One cell, centered in Poincare ball model Main cells Cells with extended edges to ideal boundary

## Symmetry

It has a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.

## Related polytopes and honeycombs

The order-5 cubic honeycomb has a related alternated honeycomb, , with icosahedron and tetrahedron cells.

The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space:

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form:

[5,3,4] family honeycombs
{5,3,4}
r{5,3,4}
t{5,3,4}
rr{5,3,4}
t0,3{5,3,4}
tr{5,3,4}
t0,1,3{5,3,4}
t0,1,2,3{5,3,4}
{4,3,5}
r{4,3,5}
t{4,3,5}
rr{4,3,5}
2t{4,3,5}
tr{4,3,5}
t0,1,3{4,3,5}
t0,1,2,3{4,3,5}

The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures.

It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.

### Rectified order-5 cubic honeycomb

Rectified order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{4,3,5} or 2r{5,3,4}
2r{5,31,1}
Coxeter diagram
Cells r{4,3}
{3,5}
Faces triangle {3}
square {4}
Vertex figure
pentagonal prism
Coxeter group ${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties Vertex-transitive, edge-transitive

The rectified order-5 cubic honeycomb, , has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.

#### Related honeycomb

It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling, r{4,5} with square and pentagonal faces

There are four rectified compact regular honeycombs:

 Image Symbols Vertexfigure r{5,3,4} r{4,3,5} r{3,5,3} r{5,3,5}
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 cubic honeycomb

Truncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{4,3,5}
Coxeter diagram
Cells t{4,3}
{3,5}
Faces triangle {3}
octagon {8}
Vertex figure
pentagonal pyramid
Coxeter group ${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
Properties Vertex-transitive

The truncated order-5 cubic honeycomb, , has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.

It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t{4,5}, with truncated square and pentagonal faces:

It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, which has octahedral cells at the truncated vertices.

#### Related honeycombs

 Image Symbols Vertexfigure t{5,3,4} t{4,3,5} t{3,5,3} t{5,3,5}

### Bitruncated order-5 cubic honeycomb

The bitruncated order-5 cubic honeycomb is the same as the bitruncated order-4 dodecahedral honeycomb.

### Cantellated order-5 cubic honeycomb

Cantellated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol rr{4,3,5}
Coxeter diagram
Cells rr{4,3}
r{3,5}
{}x{5}
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter group ${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
Properties Vertex-transitive

The cantellated order-5 cubic honeycomb, , has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.

#### Related honeycombs

It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:

### Cantitruncated order-5 cubic honeycomb

Cantitruncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{4,3,5}
Coxeter diagram
Cells tr{4,3}
t{3,5}
{}x{5}
Faces square {4}
pentagon {5}
hexagon {6}
octagon {8}
Vertex figure
mirrored sphenoid
Coxeter group ${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
Properties Vertex-transitive

The cantitruncated order-5 cubic honeycomb, , has truncated cuboctahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.

#### Related honeycombs

It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:

 Image Symbols Vertexfigure tr{5,3,4} tr{4,3,5} tr{3,5,3} tr{5,3,5}

### Runcinated order-5 cubic honeycomb

Runcinated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbol t0,3{4,3,5}
Coxeter diagram
Cells {4,3}
{5,3}
{}x{5}
Faces square {4}
pentagon {5}
Vertex figure
irregular triangular antiprism
Coxeter group ${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
Properties Vertex-transitive

The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb , has cube, dodecahedron, and pentagonal prism cells, with an irregular triangular antiprism vertex figure.

It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, with square and pentagonal faces:

#### Related honeycombs

It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:

Image Symbols Vertexfigure t0,3{4,3,5} t0,3{3,5,3} t0,3{5,3,5}

### Runcitruncated order-5 cubic honeycomb

Runctruncated order-5 cubic honeycomb
Runcicantellated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,3{4,3,5}
Coxeter diagram
Cells t{4,3}
rr{5,3}
{}x{5}
{}x{8}
Faces triangle {3}
square {4}
pentagon {5}
octagon {8}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter group ${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
Properties Vertex-transitive

The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb, , has truncated cube, rhombicosidodecahedron, pentagonal prism, and octagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

#### Related honeycombs

It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:

### Runcicantellated order-5 cubic honeycomb

The runcicantellated order-5 cubic honeycomb is the same as the runcitruncated order-4 dodecahedral honeycomb.

### Omnitruncated order-5 cubic honeycomb

Omnitruncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbol t0,1,2,3{4,3,5}
Coxeter diagram
Cells tr{5,3}
tr{4,3}
{10}x{}
{8}x{}
Faces square {4}
hexagon {6}
octagon {8}
decagon {10}
Vertex figure
irregular tetrahedron
Coxeter group ${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
Properties Vertex-transitive

The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated cuboctahedron, decagonal prism, and octagonal prism cells, with an irregular tetrahedral vertex figure.

#### Related honeycombs

It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:

### Alternated order-5 cubic honeycomb

Alternated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h{4,3,5}
Coxeter diagram
Cells {3,3}
{3,5}
Faces triangle {3}
Vertex figure
icosidodecahedron
Coxeter group ${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties Vertex-transitive, edge-transitive, quasiregular

In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). With Schläfli symbol h{4,3,5}, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.

#### Related honeycombs

It has 3 related forms: the cantic order-5 cubic honeycomb, , the runcic order-5 cubic honeycomb, , and the runcicantic order-5 cubic honeycomb, .

### Cantic order-5 cubic honeycomb

Cantic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h2{4,3,5}
Coxeter diagram
Cells r{5,3}
t{3,5}
t{3,3}
Faces triangle {3}
pentagon {5}
hexagon {6}
Vertex figure
rectangular pyramid
Coxeter group ${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties Vertex-transitive

The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2{4,3,5}. It has icosidodecahedron, truncated icosahedron, and truncated tetrahedron cells, with a rectangular pyramid vertex figure.

### Runcic order-5 cubic honeycomb

Runcic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h3{4,3,5}
Coxeter diagram
Cells {5,3}
rr{5,3}
{3,3}
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure
triangular frustum
Coxeter group ${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties Vertex-transitive

The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h3{4,3,5}. It has dodecahedron, rhombicosidodecahedron, and tetrahedron cells, with a triangular frustum vertex figure.

### Runcicantic order-5 cubic honeycomb

Runcicantic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h2,3{4,3,5}
Coxeter diagram
Cells t{5,3}
tr{5,3}
t{3,3}
Faces triangle {3}
square {4}
hexagon {6}
decagon {10}
Vertex figure
irregular tetrahedron
Coxeter group ${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties Vertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2,3{4,3,5}. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.

## 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)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
• Norman Johnson Uniform Polytopes, Manuscript
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• N.W. Johnson: Geometries and Transformations, (2015) Chapter 13: Hyperbolic Coxeter groups