Order-5 cubic honeycomb

Template:Short description

Order-5 cubic honeycomb
File:H3 435 CC center.png Poincaré disk models
Type	Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol	Template:Math
Coxeter diagram	Template:CDD
Cells	Template:Math (cube) File:Uniform polyhedron-43-t0.png
Faces	Template:Math (square)
Edge figure	Template:Math (pentagon)
Vertex figure	File:Order-5 cubic honeycomb verf.svg icosahedron
Coxeter group	Template:Math
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 Template:Math it has five cubes Template:Math around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

Template:Honeycomb

Description

Error creating thumbnail:

It is analogous to the 2D hyperbolic order-5 square tiling, {4,5}

File:Order-5 cubic honeycomb cell.png One cell, centered in Poincare ball model	Error creating thumbnail: Main cells	File:Hyperb gcubic hc.png Cells with extended edges to ideal boundary

Symmetry

It has a radical 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, Template:CDD ↔ Template:CDD, with icosahedron and tetrahedron cells.

The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space: Template:Regular compact H3 honeycombs

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form: Template:534 family

The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures. Template:Icosahedral vertex figure tessellations

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. Template:Cubic cell tessellations

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{5,3^1,1}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	r{4,3} Error creating thumbnail: {3,5} Error creating thumbnail:
Faces	triangle {3} square {4}
Vertex figure	Error creating thumbnail: pentagonal prism
Coxeter group	${\overline{B H}}_{3}$ , [4,3,5] ${\overline{D H}}_{3}$ , [5,3^1,1]
Properties	Vertex-transitive, edge-transitive

The rectified order-5 cubic honeycomb, Template:CDD, has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.

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Related honeycomb

File:H2-5-4-rectified.svg

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: Template:Rectified compact H3 honeycombs

Template:Pentagonal prism vertex figure tessellations Template:Clear

Truncated order-5 cubic honeycomb

Truncated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t{4,3,5}
Coxeter diagram	Template:CDD
Cells	t{4,3} File:Uniform polyhedron-43-t01.png {3,5} Error creating thumbnail:
Faces	triangle {3} octagon {8}
Vertex figure	Error creating thumbnail: pentagonal pyramid
Coxeter group	${\overline{B H}}_{3}$ , [4,3,5]
Properties	Vertex-transitive

The truncated order-5 cubic honeycomb, Template:CDD, has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.

File: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:

File: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.

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Related honeycombs

Template:Truncated compact H3 honeycombs

Template:Clear

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	Template:CDD
Cells	rr{4,3} File:Uniform polyhedron-43-t02.png r{3,5} File:Uniform polyhedron-53-t1.png {}x{5} File:Pentagonal prism.png
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	File:Cantellated order-5 cubic honeycomb verf.png wedge
Coxeter group	${\overline{B H}}_{3}$ , [4,3,5]
Properties	Vertex-transitive

The cantellated order-5 cubic honeycomb, Template:CDD, has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.

File:H3 534-0101 center ultrawide.png

Related honeycombs

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

File:Cantellated cubic honeycomb.png

Template:Cantellated compact H3 honeycombs Template:Clear

Cantitruncated order-5 cubic honeycomb

Cantitruncated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	tr{4,3,5}
Coxeter diagram	Template:CDD
Cells	tr{4,3} Error creating thumbnail: t{3,5} File:Uniform polyhedron-53-t12.png {}x{5} File:Pentagonal prism.png
Faces	square {4} pentagon {5} hexagon {6} octagon {8}
Vertex figure	File:Cantitruncated order-5 cubic honeycomb verf.png mirrored sphenoid
Coxeter group	${\overline{B H}}_{3}$ , [4,3,5]
Properties	Vertex-transitive

The cantitruncated order-5 cubic honeycomb, Template:CDD, has truncated cuboctahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.

File:H3 534-0111 center ultrawide.png

Related honeycombs

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

File:2-Kuboktaederstumpf 1-Oktaederstumpf 1-Hexaeder.png

Template:Cantitruncated compact H3 honeycombs Template:Clear

Runcinated order-5 cubic honeycomb

Runcinated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space Semiregular honeycomb
Schläfli symbol	t_0,3{4,3,5}
Coxeter diagram	Template:CDD
Cells	{4,3} File:Uniform polyhedron-43-t0.png {5,3} File:Uniform polyhedron-53-t0.png {}x{5} File:Pentagonal prism.png
Faces	square {4} pentagon {5}
Vertex figure	irregular triangular antiprism
Coxeter group	${\overline{B H}}_{3}$ , [4,3,5]
Properties	Vertex-transitive

The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb Template:CDD, has cube, dodecahedron, and pentagonal prism cells, with an irregular triangular antiprism vertex figure.

File:H3 534-1001 center ultrawide.png

It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, Template:CDD with square and pentagonal faces:

Error creating thumbnail:

Related honeycombs

It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t_0,3{4,3,4}:

File:Runcinated cubic honeycomb.png

Template:Runcinated compact H3 honeycombs Template:Clear

Runcitruncated order-5 cubic honeycomb

Runctruncated order-5 cubic honeycomb Runcicantellated order-4 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t_0,1,3{4,3,5}
Coxeter diagram	Template:CDD
Cells	t{4,3} File:Uniform polyhedron-43-t01.png rr{5,3} {}x{5} File:Pentagonal prism.png {}x{8} File:Octagonal prism.png
Faces	triangle {3} square {4} pentagon {5} octagon {8}
Vertex figure	File:Runcitruncated order-5 cubic honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter group	${\overline{B H}}_{3}$ , [4,3,5]
Properties	Vertex-transitive

The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb, Template:CDD, has truncated cube, rhombicosidodecahedron, pentagonal prism, and octagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

Error creating thumbnail:

Related honeycombs

It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t_0,1,3{4,3,4}:

File:Runcitruncated cubic honeycomb.jpg

Template:Runcitruncated compact H3 honeycombs Template:Clear

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	t_0,1,2,3{4,3,5}
Coxeter diagram	Template:CDD
Cells	tr{5,3} File:Uniform polyhedron-53-t012.png tr{4,3} Error creating thumbnail: {10}x{} File:Decagonal prism.png {8}x{} File:Octagonal prism.png
Faces	square {4} hexagon {6} octagon {8} decagon {10}
Vertex figure	irregular tetrahedron
Coxeter group	${\overline{B H}}_{3}$ , [4,3,5]
Properties	Vertex-transitive

The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb, Template:CDD, has truncated icosidodecahedron, truncated cuboctahedron, decagonal prism, and octagonal prism cells, with an irregular tetrahedral vertex figure.

File:H3 534-1111 center ultrawide.png

Related honeycombs

It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t_0,1,2,3{4,3,4}:

File:Omnitruncated cubic honeycomb1.png

Template:Omnitruncated compact H3 honeycombs Template:Clear

Alternated order-5 cubic honeycomb

Alternated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h{4,3,5}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	{3,3} {3,5} Error creating thumbnail:
Faces	triangle {3}
Vertex figure	File:Alternated order-5 cubic honeycomb verf.png icosidodecahedron
Coxeter group	${\overline{D H}}_{3}$ , [5,3^1,1]
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.

Error creating thumbnail: Template:Clear

Related honeycombs

Cantic order-5 cubic honeycomb

Cantic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h₂{4,3,5}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	r{5,3} File:Uniform polyhedron-53-t1.png t{3,5} File:Uniform polyhedron-53-t12.png t{3,3} File:Uniform polyhedron-33-t01.png
Faces	triangle {3} pentagon {5} hexagon {6}
Vertex figure	File:Truncated alternated order-5 cubic honeycomb verf.png rectangular pyramid
Coxeter group	${\overline{D H}}_{3}$ , [5,3^1,1]
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.

File:H3 5311-0110 center ultrawide.png Template:Clear

Runcic order-5 cubic honeycomb

Runcic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h₃{4,3,5}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	{5,3} File:Uniform polyhedron-53-t0.png rr{5,3} {3,3} Error creating thumbnail:
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	triangular frustum
Coxeter group	${\overline{D H}}_{3}$ , [5,3^1,1]
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.

Error creating thumbnail: Template:Clear

Runcicantic order-5 cubic honeycomb

Runcicantic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h_2,3{4,3,5}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	t{5,3} Error creating thumbnail: tr{5,3} File:Uniform polyhedron-53-t012.png t{3,3} File:Uniform polyhedron-33-t01.png
Faces	triangle {3} square {4} hexagon {6} decagon {10}
Vertex figure	Error creating thumbnail: irregular tetrahedron
Coxeter group	${\overline{D H}}_{3}$ , [5,3^1,1]
Properties	Vertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h_2,3{4,3,5}. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.

File:H3 5311-1110 center ultrawide.png Template:Clear

References

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Template:Isbn. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 Template:Isbn (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

Order-5 cubic honeycomb

Contents

Description

Symmetry