Order 5 dodecahedral honeycomb

Updated on Sep 21, 2024

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The order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.

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 Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°.

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

There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb, {5,3,4}, which has only four dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space (the surface of a 4-dimensional sphere), with three dodecahedra on each edge, {5,3,3}. Lastly the dodecahedral ditope, {5,3,2} exists on a 3-sphere, with 2 hemispherical cells.

There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t_1,2{5,3,5}, , of this honeycomb has all truncated icosahedron cells.

The Seifert–Weber space is a compact manifold that can be formed as a quotient space of the order-5 dodecahedral honeycomb.

This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:

This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:

Rectified order-5 dodecahedral honeycomb

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

There are four rectified compact regular honeycombs:

Truncated order-5 dodecahedral honeycomb

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

Bitruncated order-5 dodecahedral honeycomb

The bitruncated order-5 dodecahedral honeycomb, , has truncated icosahedron cells, with a disphenoid vertex figure.

Cantellated order-5 dodecahedral honeycomb

The cantellated order-5 dodecahedral honeycomb, , has alternating rhombicosidodecahedron and icosidodecahedron cells, with a triangular prism vertex figure.

Cantitruncated order-5 dodecahedral honeycomb

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

Runcinated order-5 dodecahedral honeycomb

The runcinated order-5 dodecahedral honeycomb, , has dodecahedron and pentagonal prism cells, with a triangular antiprism vertex figure.

Runcitruncated order-5 dodecahedral honeycomb

The runcitruncated order-5 dodecahedral honeycomb, , has truncated dodecahedron, icosidodecahedron and pentagonal prism cells, with a distorted square pyramid vertex figure.

Omnitruncated order-5 dodecahedral honeycomb

The omnitruncated order-5 dodecahedral honeycomb, , has truncated icosidodecahedron and decagonal prism cells, with a disphenoid vertex figure.

References

Order-5 dodecahedral honeycomb Wikipedia

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