Order-5 cubic honeycomb

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Short description: Regular tiling of hyperbolic 3-space
Order-5 cubic honeycomb
H3 435 CC center.png
Poincaré disk models
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {4,3,5}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Cells {4,3} (cube)
Uniform polyhedron-43-t0.png
Faces {4} (square)
Edge figure {5} (pentagon)
Vertex figure Order-5 cubic honeycomb verf.svg
icosahedron
Coxeter group BH3, [4,3,5]
Dual Order-4 dodecahedral honeycomb
Properties Regular

In hyperbolic geometry, 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.

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

It is analogous to the 2D hyperbolic order-5 square tiling, {4,5}
Order-5 cubic honeycomb cell.png
One cell, centered in Poincare ball model 		Hyperb gcubic hc constr.png
Main cells 		Hyperb gcubic hc.png
Cells with extended edges to ideal boundary

Symmetry

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

Related polytopes and honeycombs

The order-5 cubic honeycomb has a related alternated honeycomb, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png ↔ CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png, 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 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

Rectified order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{4,3,5} or 2r{5,3,4}
2r{5,31,1}
Coxeter diagram CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png ↔ CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 11.png
Cells r{4,3} Uniform polyhedron-43-t1.png
{3,5} Uniform polyhedron-53-t2.png
Faces triangle {3}
square {4}
Vertex figure Rectified order-5 cubic honeycomb verf.png
pentagonal prism
Coxeter group [math]\displaystyle{ \overline{BH}_3 }[/math], [4,3,5]
[math]\displaystyle{ \overline{DH}_3 }[/math], [5,31,1]
Properties Vertex-transitive, edge-transitive

The rectified order-5 cubic honeycomb, CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png, has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.

H3 435 CC center 0100.png

Related honeycomb

It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling, r{4,5} with square and pentagonal faces

There are four rectified compact regular honeycombs:

Truncated order-5 cubic honeycomb

Truncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{4,3,5}
Coxeter diagram CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Cells t{4,3} Uniform polyhedron-43-t01.png
{3,5} Uniform polyhedron-53-t2.png
Faces triangle {3}
octagon {8}
Vertex figure Truncated order-5 cubic honeycomb verf.png
pentagonal pyramid
Coxeter group [math]\displaystyle{ \overline{BH}_3 }[/math], [4,3,5]
Properties Vertex-transitive

The truncated order-5 cubic honeycomb, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png, has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.

H3 534-0011 center ultrawide.png

It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t{4,5}, with truncated square and pentagonal faces:

H2-5-4-trunc-primal.svg

It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, which has octahedral cells at the truncated vertices.

Truncated cubic honeycomb.png

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

Cantellated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol rr{4,3,5}
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Cells rr{4,3} Uniform polyhedron-43-t02.png
r{3,5} 40px
{}x{5} Pentagonal prism.png
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure Cantellated order-5 cubic honeycomb verf.png
wedge
Coxeter group [math]\displaystyle{ \overline{BH}_3 }[/math], [4,3,5]
Properties Vertex-transitive

The cantellated order-5 cubic honeycomb, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png, has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.

H3 534-0101 center ultrawide.png

Related honeycombs

It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:

Cantellated cubic honeycomb.png

Cantitruncated order-5 cubic honeycomb

Cantitruncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{4,3,5}
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Cells tr{4,3} Uniform polyhedron-43-t012.png
t{3,5} 40px
{}x{5} Pentagonal prism.png
Faces square {4}
pentagon {5}
hexagon {6}
octagon {8}
Vertex figure Cantitruncated order-5 cubic honeycomb verf.png
mirrored sphenoid
Coxeter group [math]\displaystyle{ \overline{BH}_3 }[/math], [4,3,5]
Properties Vertex-transitive

The cantitruncated order-5 cubic honeycomb, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png, has truncated cuboctahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.

H3 534-0111 center ultrawide.png

Related honeycombs

It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:

2-Kuboktaederstumpf 1-Oktaederstumpf 1-Hexaeder.png

Runcinated order-5 cubic honeycomb

Runcinated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbol t0,3{4,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Cells {4,3} Uniform polyhedron-43-t0.png
{5,3} 40px
{}x{5} Pentagonal prism.png
Faces square {4}
pentagon {5}
Vertex figure Runcinated order-5 cubic honeycomb verf.png
irregular triangular antiprism
Coxeter group [math]\displaystyle{ \overline{BH}_3 }[/math], [4,3,5]
Properties Vertex-transitive

The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png, has cube, dodecahedron, and pentagonal prism cells, with an irregular triangular antiprism vertex figure.

H3 534-1001 center ultrawide.png

It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png with square and pentagonal faces:

H2-5-4-cantellated.svg

Related honeycombs

It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:

Runcinated cubic honeycomb.png

Runcitruncated order-5 cubic honeycomb

Runctruncated order-5 cubic honeycomb
Runcicantellated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,3{4,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Cells t{4,3} Uniform polyhedron-43-t01.png
rr{5,3} 40px
{}x{5} 40px
{}x{8} Octagonal prism.png
Faces triangle {3}
square {4}
pentagon {5}
octagon {8}
Vertex figure Runcitruncated order-5 cubic honeycomb verf.png
isosceles-trapezoidal pyramid
Coxeter group [math]\displaystyle{ \overline{BH}_3 }[/math], [4,3,5]
Properties Vertex-transitive

The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png, has truncated cube, rhombicosidodecahedron, pentagonal prism, and octagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

H3 534-1011 center ultrawide.png

Related honeycombs

It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:

Runcitruncated cubic honeycomb.jpg

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

Omnitruncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbol t0,1,2,3{4,3,5}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Cells tr{5,3} Uniform polyhedron-53-t012.png
tr{4,3} 40px
{10}x{} 40px
{8}x{} Octagonal prism.png
Faces square {4}
hexagon {6}
octagon {8}
decagon {10}
Vertex figure Omnitruncated order-4 dodecahedral honeycomb verf.png
irregular tetrahedron
Coxeter group [math]\displaystyle{ \overline{BH}_3 }[/math], [4,3,5]
Properties Vertex-transitive

The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb, CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png, has truncated icosidodecahedron, truncated cuboctahedron, decagonal prism, and octagonal prism cells, with an irregular tetrahedral vertex figure.

H3 534-1111 center ultrawide.png

Related honeycombs

It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:

Omnitruncated cubic honeycomb1.png

Alternated order-5 cubic honeycomb

Alternated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h{4,3,5}
Coxeter diagram CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png ↔ CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
Cells {3,3} Uniform polyhedron-33-t0.png
{3,5} Uniform polyhedron-53-t2.png
Faces triangle {3}
Vertex figure Alternated order-5 cubic honeycomb verf.png
icosidodecahedron
Coxeter group [math]\displaystyle{ \overline{DH}_3 }[/math], [5,31,1]
Properties Vertex-transitive, edge-transitive, quasiregular

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.

Alternated order 5 cubic honeycomb.png

Related honeycombs

It has 3 related forms: the cantic order-5 cubic honeycomb, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png, the runcic order-5 cubic honeycomb, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png, and the runcicantic order-5 cubic honeycomb, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png.

Cantic order-5 cubic honeycomb

Cantic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h2{4,3,5}
Coxeter diagram CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png ↔ CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node.png
Cells r{5,3} Uniform polyhedron-53-t1.png
t{3,5} 40px
t{3,3} Uniform polyhedron-33-t01.png
Faces triangle {3}
pentagon {5}
hexagon {6}
Vertex figure Truncated alternated order-5 cubic honeycomb verf.png
rectangular pyramid
Coxeter group [math]\displaystyle{ \overline{DH}_3 }[/math], [5,31,1]
Properties Vertex-transitive

The cantic 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.

H3 5311-0110 center ultrawide.png

Runcic order-5 cubic honeycomb

Runcic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h3{4,3,5}
Coxeter diagram CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png ↔ CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node 1.png
Cells {5,3} Uniform polyhedron-53-t0.png
rr{5,3} 40px
{3,3} Uniform polyhedron-33-t0.png
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure Runcinated alternated order-5 cubic honeycomb verf.png
triangular frustum
Coxeter group [math]\displaystyle{ \overline{DH}_3 }[/math], [5,31,1]
Properties Vertex-transitive

The runcic 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.

H3 5311-1010 center ultrawide.png

Runcicantic order-5 cubic honeycomb

Runcicantic order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol h2,3{4,3,5}
Coxeter diagram CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png ↔ CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 5.pngCDel node 1.png
Cells t{5,3} Uniform polyhedron-53-t01.png
tr{5,3} 40px
t{3,3} Uniform polyhedron-33-t01.png
Faces triangle {3}
square {4}
hexagon {6}
decagon {10}
Vertex figure Runcitruncated alternated order-5 cubic honeycomb verf.png
irregular tetrahedron
Coxeter group [math]\displaystyle{ \overline{DH}_3 }[/math], [5,31,1]
Properties Vertex-transitive

The runcicantic 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.

H3 5311-1110 center ultrawide.png

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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