Order-5 cubic honeycomb

In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol ${4,3,5}$ it has five cubes ${4,3}$ around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

Symmetry
It has a radical subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.

Related polytopes and honeycombs
The order-5 cubic honeycomb has a related alternated honeycomb, ↔, with icosahedron and tetrahedron cells.

The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space:

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form:

The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures.

It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.

Rectified order-5 cubic honeycomb
The rectified order-5 cubic honeycomb,, has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.



Related honeycomb


There are four rectified compact regular honeycombs:

Truncated order-5 cubic honeycomb
The truncated order-5 cubic honeycomb,, has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.



It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t{4,5}, with truncated square and pentagonal faces:
 * H2-5-4-trunc-primal.svg

It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, which has octahedral cells at the truncated vertices.
 * Truncated_cubic_honeycomb.png

Bitruncated order-5 cubic honeycomb
The bitruncated order-5 cubic honeycomb is the same as the bitruncated order-4 dodecahedral honeycomb.

Cantellated order-5 cubic honeycomb
The cantellated order-5 cubic honeycomb,, has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.



Related honeycombs
It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:
 * Cantellated cubic honeycomb.png

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



Related honeycombs
It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:
 * 2-Kuboktaederstumpf 1-Oktaederstumpf 1-Hexaeder.png

Runcinated order-5 cubic honeycomb
The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb, has cube, dodecahedron, and pentagonal prism cells, with an irregular triangular antiprism vertex figure.



It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, with square and pentagonal faces:
 * H2-5-4-cantellated.svg

Related honeycombs
It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:
 * Runcinated cubic honeycomb.png

Runcitruncated order-5 cubic honeycomb
The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb,, has truncated cube, rhombicosidodecahedron, pentagonal prism, and octagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.



Related honeycombs
It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:
 * Runcitruncated cubic honeycomb.jpg

Runcicantellated order-5 cubic honeycomb
The runcicantellated order-5 cubic honeycomb is the same as the runcitruncated order-4 dodecahedral honeycomb.

Omnitruncated order-5 cubic honeycomb
The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb,, has truncated icosidodecahedron, truncated cuboctahedron, decagonal prism, and octagonal prism cells, with an irregular tetrahedral vertex figure.



Related honeycombs
It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:
 * Omnitruncated cubic honeycomb1.png

Alternated order-5 cubic honeycomb
In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). With Schläfli symbol h{4,3,5}, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.



Related honeycombs
It has 3 related forms: the cantic order-5 cubic honeycomb,, the runcic order-5 cubic honeycomb, , and the runcicantic order-5 cubic honeycomb,.

Cantic order-5 cubic honeycomb
The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2{4,3,5}. It has icosidodecahedron, truncated icosahedron, and truncated tetrahedron cells, with a rectangular pyramid vertex figure.



Runcic order-5 cubic honeycomb
The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h3{4,3,5}. It has dodecahedron, rhombicosidodecahedron, and tetrahedron cells, with a triangular frustum vertex figure.



Runcicantic order-5 cubic honeycomb
The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2,3{4,3,5}. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.