# 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 | triangle {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:
* Centered on dodecahedron
* Centered on icosahedron

## 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 | triangle {3} square {4} pentagon {5} |
| 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 | triangle {3} decagon {10} |
| Vertex figure | pentagonal 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 and icosahedron cells, in a pentagonal 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 | pentagon {5} 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 and truncated icosahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram .
    Perspective view from center of dodecahedron
It can be seen as somewhat analogous to the pentahexagonal tiling, which has 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 | triangle {3} square {4} pentagon {5} 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, and 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