Order-5 cubic honeycomb


The order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol, it has five cubes around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

Description


One cell, centered in Poincare ball model
Main cells
Cells with extended edges to ideal boundary

Symmetry

It has a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: , 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 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, with truncated square and pentagonal faces:
It is similar to the Euclidean truncated cubic honeycomb, t, which has octahedral cells at the truncated vertices.

Related honeycombs

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 cantellated cubic honeycomb, rr:

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 cantitruncated cubic honeycomb, tr:

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, with square and pentagonal faces:

Related honeycombs

It is similar to the Euclidean runcinated cubic honeycomb, t0,3:

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 runcitruncated cubic honeycomb, t0,1,3:

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 omnitruncated cubic honeycomb, t0,1,2,3:

Alternated order-5 cubic honeycomb

In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation. With Schläfli symbol h, 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, with Schläfli symbol h2. 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, with Schläfli symbol h3. 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, with Schläfli symbol h2,3. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.