In hyperbolic geometry, 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.

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, {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, t1,2{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 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