Order-5 dodecahedral honeycomb


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

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°.

Images

Related polytopes and honeycombs

There are four regular compact honeycombs in 3D hyperbolic space:
There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb,, 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, with three dodecahedra on each edge,. Lastly the dodecahedral ditope, exists on a 3-sphere, with 2 hemispherical cells.
There are nine uniform honeycombs in the Coxeter group family, including this regular form. Also the bitruncated form, t1,2,, 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 regular polytopes 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.

Related tilings and honeycomb

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.

Related honeycombs

Bitruncated order-5 dodecahedral honeycomb

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

Related honeycombs

Cantellated order-5 dodecahedral honeycomb

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

Related honeycombs

Cantitruncated order-5 dodecahedral honeycomb

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

Related honeycombs

Runcinated order-5 dodecahedral honeycomb

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

Related honeycombs

Runcitruncated order-5 dodecahedral honeycomb

The runcitruncated order-5 dodecahedral honeycomb,, has truncated dodecahedron, rhombicosidodecahedron, pentagonal prism, and decagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.
The runcicantellated order-5 dodecahedral honeycomb is equivalent to the runcitruncated order-5 dodecahedral honeycomb.

Related honeycombs

Omnitruncated order-5 dodecahedral honeycomb

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

Related honeycombs