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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.
Contents
- Symmetry
- Related polytopes and honeycombs
- Compact regular honeycombs
- 543 honeycombs
- Polytopes with icosahedral vertex figures
- Related polytopes and honeycombs with cubic cells
- Rectified order 5 cubic honeycomb
- Related honeycomb
- Truncated order 5 cubic honeycomb
- Bitruncated order 5 cubic honeycomb
- Cantellated order 5 cubic honeycomb
- Related honeycombs
- Cantitruncated order 5 cubic honeycomb
- Runcinated order 5 cubic honeycomb
- Runcitruncated order 5 cubic honeycomb
- Omnitruncated order 5 cubic honeycomb
- Alternated order 5 cubic honeycomb
- Cantic order 5 cubic honeycomb
- Runcic order 5 cubic honeycomb
- Runcicantic order 5 cubic honeycomb
- References
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Symmetry
It a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.
Related polytopes and honeycombs
It has a related alternation honeycomb, represented by ↔ , having icosahedron and tetrahedron cells.
Compact regular honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
543 honeycombs
There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form:
Polytopes with icosahedral vertex figures
It is in a sequence of polychora and honeycomb with icosahedron vertex figures:
Related polytopes and honeycombs with cubic cells
It 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:
It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, with octahedral cells at the truncated vertices.
Bitruncated order-5 cubic honeycomb
Same as Bitruncated order-4 dodecahedral honeycomb
Cantellated order-5 cubic honeycomb
The cantellated order-5 cubic honeycomb, , has rhombicuboctahedron and icosidodecahedron cells, with a wedge vertex figure.
Related honeycombs
It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:
Cantitruncated order-5 cubic honeycomb
The cantitruncated order-5 cubic honeycomb, , has rhombicuboctahedron and icosidodecahedron cells, with a mirrored sphenoid vertex figure.
Related honeycombs
It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:
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 octahedron vertex figure.
It is analogous the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, with square and pentagonal faces:
Related honeycombs
It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:
Runcitruncated order-5 cubic honeycomb
The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb , has cube, dodecahedron, and pentagonal prism cells, with a quad-pyramid vertex figure.
Related honeycombs
It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:
Omnitruncated order-5 cubic honeycomb
The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb has Coxeter diagram .
Related honeycombs
It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:
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). It has Schläfli symbol h2{4,3,5} and 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). It has Schläfli symbol h3{4,3,5} and a triangular prism vertex figure.
Runcicantic order-5 cubic honeycomb
The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h2,3{4,3,5} and a mirrored sphenoid vertex figure.