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In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs) in Hyperbolic space, hyperbolic 3-space. With Schläfli symbol it has five cubes around each Edge (geometry), edge, and 20 cubes around each Vertex (geometry), vertex. It is Dual polytope, dual with the order-4 dodecahedral honeycomb.
There are four rectified compact regular honeycombs:
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 (order-4) truncated cubic honeycomb, t, which has octahedral cells at the truncated vertices.
:
It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr, with square and pentagonal faces:
: 
Description
Symmetry
It has a radial subgroup symmetry construction with regular dodecahedron, dodecahedral fundamental domains: Coxeter_notation#Extended_symmetry, Coxeter notation: [4,(3,5)*], index 120.Related polytopes and honeycombs
The order-5 cubic honeycomb has a related Alternation (geometry), alternated honeycomb, ↔ , with regular icosahedron, icosahedron and tetrahedron cells. The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space: There are Convex_uniform_honeycombs_in_hyperbolic_space#.5B5.2C3.2C4.5D_family, 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 regular icosahedron, icosahedral vertex figures. It is also in a sequence of regular polychora and honeycombs with cube, cubic cell (geometry), 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 regular icosahedron, 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 regular icosahedron, 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 (order-4) 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 (geometry), wedge vertex figure.Related honeycombs
It is similar to the Euclidean (order-4) 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 (order-4) cantitruncated cubic honeycomb, tr: :
Runcinated order-5 cubic honeycomb
The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb , has cube, regular dodecahedron, 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 (order-4) 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 trapezoid, isosceles-trapezoidal pyramid (geometry), pyramid vertex figure.Related honeycombs
It is similar to the Euclidean (order-4) 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 tetrahedron, tetrahedral vertex figure.
Related honeycombs
It is similar to the Euclidean (order-4) 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 (or honeycomb (geometry), honeycomb). With Schläfli symbol h, it can be considered a quasiregular honeycomb, alternating regular icosahedron, icosahedra and tetrahedron, 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 (geometry), honeycomb), with Schläfli symbol h2. It has icosidodecahedron, truncated icosahedron, and truncated tetrahedron cells, with a rectangular pyramid (geometry), pyramid vertex figure.
Runcic order-5 cubic honeycomb
The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb (geometry), honeycomb), with Schläfli symbol h3. It has regular dodecahedron, 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 (geometry), honeycomb), with Schläfli symbol h2,3. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.
See also
* Convex uniform honeycombs in hyperbolic space * List_of_regular_polytopes#Tessellations_of_hyperbolic_3-space, Regular tessellations of hyperbolic 3-spaceReferences
*H.S.M. Coxeter, Coxeter, ''Regular Polytopes (book), Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294-296) *H.S.M. Coxeter, Coxeter, ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 {{isbn, 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213) * Norman Johnson (mathematician), Norman Johnson ''Uniform Polytopes'', Manuscript ** Norman Johnson (mathematician), N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 ** N.W. Johnson: ''Geometries and Transformations'', (2015) Chapter 13: Hyperbolic Coxeter groups Honeycombs (geometry)