# Dodecahedral-icosahedral honeycomb

Dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol {(3,5,3,5)} or {(5,3,5,3)}
Coxeter diagram or
Cells {5,3}
{3,5}
r{5,3}
Faces triangular {3}
pentagon {5}
Vertex figure
rhombicosidodecahedron
Coxeter group [(5,3)[2]]
Properties Vertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the dodecahedral-icosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron 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.

## Images

Wide-angle perspective views:

## Related honeycombs

There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group : , , , , .

### Rectified dodecahedral-icosahedral honeycomb

Rectified dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol r{(5,3,5,3)}
Coxeter diagrams or
Cells r{5,3}
rr{3,5}
Faces triangular {3}
decagon {10}
Vertex figure
cuboid
Coxeter group [[(5,3)[2]]],
Properties Vertex-transitive, edge-transitive

The rectified dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosidodecahedron, and rhombicosidodecahedron cells, in a cuboid vertex figure. It has a Coxeter diagram .

Perspective view from center of rhombicosidodecahedron

### Cyclotruncated dodecahedral-icosahedral honeycomb

Cyclotruncated dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol ct{(5,3,5,3)}
Coxeter diagrams or
Cells t{5,3}
{3,5}
Faces triangular {3}
decagon {10}
Vertex figure
square antiprism
Coxeter group [[(5,3)[2]]],
Properties Vertex-transitive, edge-transitive

The cyclotruncated dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from truncated dodecahedron, icosahedron cells, in a square antiprism vertex figure. It has a Coxeter diagram .

Perspective view from center of icosahedron

### Cyclotruncated icosahedral-dodecahedral honeycomb

Cyclotruncated icosahedral-dodecahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol ct{(3,5,3,5)}
Coxeter diagrams or
Cells {5,3}
t{3,5}
Faces triangular {3}
square {4}
hexagon {6}
Vertex figure
triangular antiprism
Coxeter group [[(5,3)[2]]],
Properties Vertex-transitive, edge-transitive

The cyclotruncated icosahedral-dodecahedral honeycomb is a compact uniform honeycomb, constructed from dodecahedron, truncated icosahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram .

Perspective view from center of dodecahedron

It can be see as somewhat analogous to the pentahexagonal tiling with pentagonal and hexagonal faces:

### Truncated dodecahedral-icosahedral honeycomb

Truncated dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol t{(5,3,5,3)}
Coxeter diagrams or or
or
Cells t{3,5}
t{5,3}
rr{3,5}
tr{5,3}
Faces triangular {3}
square {4}
hexagon {6}
decagon {10}
Vertex figure
trapezoidal pyramid
Coxeter group [(5,3)[2]]
Properties Vertex-transitive

The truncated dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from truncated icosahedron, truncated dodecahedron, rhombicosidodecahedron, truncated icosidodecahedron cells, in a trapezoidal pyramid vertex figure. It has a Coxeter diagram .

Perspective view from center of truncated icosahedron

### Omnitruncated dodecahedral-icosahedral honeycomb

Omnitruncated dodecahedral-icosahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol tr{(5,3,5,3)}
Coxeter diagrams
Cells tr{3,5}
Faces square {4}
hexagon {6}
decagon {10}
Vertex figure
Rhombic disphenoid
Coxeter group [(2,2)+[(5,3)[2]]],
Properties Vertex-transitive, edge-transitive, cell-transitive

The omnitruncated dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from truncated icosidodecahedron cells, in a rhombic disphenoid vertex figure. It has a Coxeter diagram .

Perspective view from center of truncated icosidodecahedron