• Source: Order-5 cubic honeycomb

In hyperbolic geometry, the 5.180.24.3/info/order" target="_blank">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 5.180.24.3/info/order" target="_blank">order-4 dodecahedral honeycomb.
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.




Description






Symmetry


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




Related polytopes and honeycombs


The 5.180.24.3/info/order" target="_blank">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 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb as the regular form:

The 5.180.24.3/info/order" target="_blank">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 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The rectified 5.180.24.3/info/order" target="_blank">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 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The truncated 5.180.24.3/info/order" target="_blank">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 5.180.24.3/info/order" target="_blank">order-5 square tiling, t{4,5}, with truncated square and pentagonal faces:

It is similar to the Euclidean (5.180.24.3/info/order" target="_blank">order-4) truncated cubic honeycomb, t{4,3,4}, which has octahedral cells at the truncated vertices.



Related honeycombs





= Bitruncated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=
The bitruncated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb is the same as the bitruncated 5.180.24.3/info/order" target="_blank">order-4 dodecahedral honeycomb.




= Cantellated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The cantellated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb, , has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.



Related honeycombs

It is similar to the Euclidean (5.180.24.3/info/order" target="_blank">order-4) cantellated cubic honeycomb, rr{4,3,4}:




= Cantitruncated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The cantitruncated 5.180.24.3/info/order" target="_blank">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 (5.180.24.3/info/order" target="_blank">order-4) cantitruncated cubic honeycomb, tr{4,3,4}:




= Runcinated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The runcinated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb or runcinated 5.180.24.3/info/order" target="_blank">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:



Related honeycombs

It is similar to the Euclidean (5.180.24.3/info/order" target="_blank">order-4) runcinated cubic honeycomb, t0,3{4,3,4}:




= Runcitruncated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The runcitruncated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb or runcicantellated 5.180.24.3/info/order" target="_blank">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 (5.180.24.3/info/order" target="_blank">order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:




= Runcicantellated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=
The runcicantellated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb is the same as the runcitruncated 5.180.24.3/info/order" target="_blank">order-4 dodecahedral honeycomb.




= Omnitruncated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The omnitruncated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb or omnitruncated 5.180.24.3/info/order" target="_blank">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 (5.180.24.3/info/order" target="_blank">order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:




= Alternated 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

In 3-dimensional hyperbolic geometry, the alternated 5.180.24.3/info/order" target="_blank">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 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb, , the runcic 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb, , and the runcicantic 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb, .




= Cantic 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The cantic 5.180.24.3/info/order" target="_blank">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 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The runcic 5.180.24.3/info/order" target="_blank">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 5.180.24.3/info/order" target="_blank">order-5 cubic honeycomb

=

The runcicantic 5.180.24.3/info/order" target="_blank">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.




See also


Convex uniform honeycombs in hyperbolic space
Regular tessellations of hyperbolic 3-space




References


Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
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 Uniform Polytopes, Manuscript
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

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