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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.
Contents
- Images
- Related honeycombs
- Rectified dodecahedral icosahedral honeycomb
- Cyclotruncated dodecahedral icosahedral honeycomb
- Cyclotruncated icosahedral dodecahedral honeycomb
- Truncated dodecahedral icosahedral honeycomb
- Omnitruncated dodecahedral icosahedral honeycomb
- References
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
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 rhombicosidodecahedronCyclotruncated dodecahedral-icosahedral honeycomb
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 icosahedronCyclotruncated icosahedral-dodecahedral honeycomb
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 dodecahedronIt can be see as somewhat analogous to the pentahexagonal tiling with pentagonal and hexagonal faces:
Truncated dodecahedral-icosahedral honeycomb
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 icosahedronOmnitruncated dodecahedral-icosahedral honeycomb
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