# Icosahedral honeycomb

Icosahedral honeycomb

Poincaré disk model
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {3,5,3}
Coxeter diagram
Cells {5,3} (regular icosahedron)
Faces {3} (triangle)
Edge figure {3} (triangle)
Vertex figure
dodecahedron
Dual Self-dual
Coxeter group J3, [3,5,3]
Properties Regular

In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

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

The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.

## Related regular honeycombs

There are four regular compact honeycombs in 3D hyperbolic space:

## Related regular polytopes and honeycombs

It is a member of a sequence of regular polychora and honeycombs {3,p,3} with deltrahedral cells:

{3,p,3} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
{3,p,3} {3,3,3} {3,4,3} {3,5,3} {3,6,3} {3,7,3} {3,8,3} ... {3,∞,3}
Image
Cells
{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,∞}
Vertex
figure

{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{∞,3}

It is also a member of a sequence of regular polychora and honeycombs {p,5,p}, with vertex figures composed of pentagons:

{p,5,p} regular honeycombs
Space H3
Form Compact Noncompact
Name {3,5,3} {4,5,4} {5,5,5} {6,5,6} {7,5,7} {8,5,8} ...{∞,5,∞}
Image
Cells
{p,5}

{3,5}

{4,5}

{5,5}

{6,5}

{7,5}

{8,5}

{∞,5}
Vertex
figure
{5,p}

{5,3}

{5,4}

{5,5}

{5,6}

{5,7}

{5,8}

{5,∞}

## Uniform honeycombs

There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.

[3,5,3] family honeycombs
{3,5,3}
t1{3,5,3}
t0,1{3,5,3}
t0,2{3,5,3}
t0,3{3,5,3}
t1,2{3,5,3}
t0,1,2{3,5,3}
t0,1,3{3,5,3}
t0,1,2,3{3,5,3}

### Rectified icosahedral honeycomb

Rectified icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{3,5,3} or t1{3,5,3}
Coxeter diagram
Cells r{3,5}
{5,3}
Faces triangle {3}
pentagon {5}
Vertex figure
triangular prism
Coxeter group ${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties Vertex-transitive, edge-transitive

The rectified icosahedral honeycomb, t1{3,5,3}, , has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:

Perspective projections from center of Poincaré disk model

#### Related honeycomb

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}

### Truncated icosahedral honeycomb

Truncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{3,5,3} or t0,1{3,5,3}
Coxeter diagram
Cells t{3,5}
{5,3}
Faces pentagon {5}
hexagon {6}
Vertex figure
triangular pyramid
Coxeter group ${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties Vertex-transitive

The truncated icosahedral honeycomb, t0,1{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.

#### Related honeycombs

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

### Bitruncated icosahedral honeycomb

Bitruncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol 2t{3,5,3} or t1,2{3,5,3}
Coxeter diagram
Cells t{5,3}
Faces triangle {3}
decagon {10}
Vertex figure
tetragonal disphenoid
Coxeter group ${\displaystyle 2\times {\overline {J}}_{3}}$, [[3,5,3]]
Properties Vertex-transitive, edge-transitive, cell-transitive

The bitruncated icosahedral honeycomb, t1,2{3,5,3}, , has truncated dodecahedron cells with a tetragonal disphenoid vertex figure.

#### Related honeycombs

Image Symbols Vertexfigure 2t{4,3,5} 2t{3,5,3} 2t{5,3,5}

### Cantellated icosahedral honeycomb

Cantellated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol rr{3,5,3} or t0,2{3,5,3}
Coxeter diagram
Cells rr{3,5}
r{5,3}
{}x{3}
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter group ${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties Vertex-transitive

The cantellated icosahedral honeycomb, t0,2{3,5,3}, , has rhombicosidodecahedron, icosidodecahedron, and triangular prism cells, with a wedge vertex figure.

#### Related honeycombs

Four cantellated regular compact honeycombs in H3
 Image Symbols Vertexfigure rr{5,3,4} rr{4,3,5} rr{3,5,3} rr{5,3,5}

### Cantitruncated icosahedral honeycomb

Cantitruncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{3,5,3} or t0,1,2{3,5,3}
Coxeter diagram
Cells tr{3,5}
t{5,3}
{}x{3}
Faces triangle {3}
square {4}
hexagon {6}
decagon {10}
Vertex figure
mirrored sphenoid
Coxeter group ${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties Vertex-transitive

The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3}, , has truncated icosidodecahedron, truncated dodecahedron, and triangular prism cells, with a mirrored sphenoid vertex figure.

#### Related honeycombs

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

### Runcinated icosahedral honeycomb

Runcinated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,3{3,5,3}
Coxeter diagram
Cells {3,5}
{}×{3}
Faces triangle {3}
square {4}
Vertex figure
pentagonal antiprism
Coxeter group ${\displaystyle 2\times {\overline {J}}_{3}}$, [[3,5,3]]
Properties Vertex-transitive, edge-transitive

The runcinated icosahedral honeycomb, t0,3{3,5,3}, , has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.

Viewed from center of triangular prism

#### Related honeycombs

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

### Runcitruncated icosahedral honeycomb

Runcitruncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,3{3,5,3}
Coxeter diagram
Cells t{3,5}
rr{3,5}
{}×{3}
{}×{6}
Faces triangle {3}
square {4}
pentagon {5}
hexagon {6}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter group ${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties Vertex-transitive

The runcitruncated icosahedral honeycomb, t0,1,3{3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism, and triangular prism cells, with an isosceles-trapezoidal pyramid vertex figure.

The runcicantellated icosahedral honeycomb is equivalent to the runcitruncated icosahedral honeycomb.

Viewed from center of triangular prism

#### Related honeycombs

Four runcitruncated regular compact honeycombs in H3
 Image Symbols Vertexfigure t0,1,3{5,3,4} t0,1,3{4,3,5} t0,1,3{3,5,3} t0,1,3{5,3,5}

### Omnitruncated icosahedral honeycomb

Omnitruncated icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,2,3{3,5,3}
Coxeter diagram
Cells tr{3,5}
{}×{6}
Faces square {4}
hexagon {6}
dodecagon {10}
Vertex figure
phyllic disphenoid
Coxeter group ${\displaystyle 2\times {\overline {J}}_{3}}$, [[3,5,3]]
Properties Vertex-transitive

The omnitruncated icosahedral honeycomb, t0,1,2,3{3,5,3}, , has truncated icosidodecahedron and hexagonal prism cells, with a phyllic disphenoid vertex figure.

Centered on hexagonal prism

#### Related honeycombs

Three omnitruncated regular compact honeycombs in H3
Image Symbols Vertexfigure t0,1,2,3{4,3,5} t0,1,2,3{3,5,3} t0,1,2,3{5,3,5}

### Omnisnub icosahedral honeycomb

Omnisnub icosahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h(t0,1,2,3{3,5,3})
Coxeter diagram
Cells sr{3,5}
s{2,3}
irr. {3,3}
Faces triangle {3}
pentagon {5}
Vertex figure
Coxeter group [[3,5,3]]+
Properties Vertex-transitive

The omnisnub icosahedral honeycomb, h(t0,1,2,3{3,5,3}), , has snub dodecahedron, octahedron, and tetrahedron cells, with an irregular vertex figure. It is vertex-transitive, but cannot be made with uniform cells.

### Partially diminished icosahedral honeycomb

Partially diminished icosahedral honeycomb
Parabidiminished icosahedral honeycomb
Type Uniform honeycombs
Schläfli symbol pd{3,5,3}
Coxeter diagram -
Cells {5,3}
s{2,5}
Faces triangle {3}
pentagon {5}
Vertex figure
tetrahedrally diminished
dodecahedron
Coxeter group 1/5[3,5,3]+
Properties Vertex-transitive

The partially diminished icosahedral honeycomb or parabidiminished icosahedral honeycomb, pd{3,5,3}, is a non-Wythoffian uniform honeycomb with dodecahedron and pentagonal antiprism cells, with a tetrahedrally diminished dodecahedron vertex figure. The icosahedral cells of the {3,5,3} are diminished at opposite vertices (parabidiminished), leaving a pentagonal antiprism (parabidiminished icosahedron) core, and creating new dodecahedron cells above and below.[1][2]