Order-5 cubic honeycomb
In hyperbolic geometry, 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 dodecahedral honeycomb.
Description
One cell, centered in Poincare ball model | Main cells | Cells with extended edges to ideal boundary |
Symmetry
It has a radical 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.
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: