Order-5 cubic honeycomb

Order-5 cubic honeycomb
Poincaré disk models
Type	Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol	{4,3,5}
Coxeter diagram
Cells	{4,3}
Faces	square {4}
Edge figure	pentagon {5}
Vertex figure	80px icosahedron
Coxeter group	BH3, [5,3,4]
Dual	Order-4 dodecahedral honeycomb
Properties	Regular

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}

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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:

Four regular compact honeycombs in H³

{5,3,4}

{4,3,5}

{3,5,3}

{5,3,5}

543 honeycombs

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form:

[5,3,4] family honeycombs

{5,3,4}	r{5,3,4}	t{5,3,4}	rr{5,3,4}	t0,3{5,3,4}	tr{5,3,4}	t0,1,3{5,3,4}	t0,1,2,3{5,3,4}
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{4,3,5}	r{4,3,5}	t{4,3,5}	rr{4,3,5}	2t{4,3,5}	tr{4,3,5}	t0,1,3{4,3,5}	t0,1,2,3{4,3,5}

Polytopes with icosahedral vertex figures

It is in a sequence of polychora and honeycomb with icosahedron vertex figures:

{p,3,5} polytopes
Space S3 H3 Form Finite Compact Paracompact Noncompact Name {3,3,5} {4,3,5} {5,3,5} {6,3,5} {7,3,5} ... {∞,3,5} Image Cells {3,3} {4,3} {5,3} {6,3} {∞,3} {∞,3}

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.

{4,3,p} regular honeycombs Space S3 E3 H3 Form Finite Affine Compact Paracompact Noncompact Name {4,3,3} {4,3,4} {4,3,5} {4,3,6} {4,3,7} {4,3,8} ... {4,3,∞} Image Vertex figure {3,3} {3,4} 60px {3,5} {3,6} {3,7} {3,8} {3,∞}

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	↔
Cells	r{4,3} {3,5}
Faces	triangle {3} square {4}
Vertex figure	80px pentagonal prism
Coxeter group	BH3, [5,3,4] DH3, [5,31,1]
Properties	Vertex-transitive, edge-transitive

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

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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:

Four rectified regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	r{5,3,4}	r{4,3,5}	r{3,5,3}	r{5,3,5}
Vertex figure	100px	100px	100px	100px

r{p,3,5}

Space	S3	H3
Form	Finite	Compact		Paracompact	Noncompact
Name	r{3,3,5}	r{4,3,5}	r{5,3,5}	r{6,3,5}	r{7,3,5}	... r{∞,3,5}
Image	60px	80px	80px	80px
Cells {3,5}	r{3,3}	r{4,3}	r{5,3}	r{6,3}	r{7,3}	r{∞,3}

Truncated order-5 cubic honeycomb

Truncated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t{4,3,5}
Coxeter diagram
Cells	t{4,3} {3,5}
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	80px pentagonal pyramid
Coxeter group	BH3, [5,3,4]
Properties	Vertex-transitive

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

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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.

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Related honeycombs

Four truncated regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	t{5,3,4}	t{4,3,5}	t{3,5,3}	t{5,3,5}
Vertex figure	100px	100px	100px	100px

Bitruncated order-5 cubic honeycomb

Same as 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
Cells	rr{4,3} r{3,5} {}x{5}
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	80px wedge
Coxeter group	BH3, [5,3,4]
Properties	Vertex-transitive

The cantellated order-5 cubic honeycomb, , has rhombicuboctahedron and icosidodecahedron cells, with a wedge vertex figure.

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Related honeycombs

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

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Cantitruncated order-5 cubic honeycomb

Cantitruncated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	tr{4,3,5}
Coxeter diagram
Cells	tr{4,3} t{3,5}
Faces	square {4} pentagon {5} hexagon {6} octahedron {8}
Vertex figure	80px Mirrored sphenoid
Coxeter group	BH3, [5,3,4]
Properties	Vertex-transitive

The cantitruncated order-5 cubic honeycomb, , has rhombicuboctahedron and icosidodecahedron cells, with a mirrored sphenoid vertex figure.

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Related honeycombs

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

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Four cantitruncated regular compact honeycombs in H³

Image	100px	100px	100px	100px
Symbols	tr{5,3,4}	tr{4,3,5}	tr{3,5,3}	tr{5,3,5}
Vertex figure	100px	100px	100px	100px

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
Cells	{4,3} {5,3} {}x{5}
Faces	Square {4} Pentagon {5}
Vertex figure	80px octahedron
Coxeter group	BH3, [5,3,4]
Properties	Vertex-transitive

The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb , has cube, dodecahedron, and pentagonal prism cells, with an octahedron vertex figure.

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

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Related honeycombs

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

Three runcinated regular compact honeycombs in H³

Image	100px	100px	100px
Symbols	t0,3{4,3,5}	t0,3{3,5,3}	t0,3{5,3,5}
Vertex figure	100px

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
Cells	t{4,3} rr{5,3} {}x{5} {}x{8}
Faces	Triangle {3} Square {4} Pentagon {5} Octagon {8}
Vertex figure	80px quad-pyramid
Coxeter group	BH3, [5,3,4]
Properties	Vertex-transitive

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.

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Related honeycombs

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

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
Cells	tr{5,3} tr{4,3} {10}x{} {8}x{}
Faces	Square {4} Hexagon {6} Octagon {8} Decagon {10}
Vertex figure	80px tetrahedron
Coxeter group	BH3, [5,3,4]
Properties	Vertex-transitive

The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb has Coxeter diagram .

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Related honeycombs

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

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Alternated order-5 cubic honeycomb

Alternated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h{4,3,5}
Coxeter diagram	↔
Cells	{3,3} {3,5}
Faces	triangle {3} pentagon {5}
Vertex figure	80px icosidodecahedron
Coxeter group	DH3, [5,31,1]
Properties	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.

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

Cantic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h2{4,3,5}
Coxeter diagram	↔
Cells	r{5,3} t{3,5} t{3,3}
Faces	Triangle {3} Pentagon {5} Hexagon {6}
Vertex figure	60px Rectangular pyramid
Coxeter group	DH3, [5,31,1]
Properties	Vertex-transitive

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.

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Runcic order-5 cubic honeycomb

Runcic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h3{4,3,5}
Coxeter diagram	↔
Cells	{5,3} rr{5,3} {3,3}
Faces	Triangle {3} square {4} pentagon {5}
Vertex figure	60px triangular prism
Coxeter group	DH3, [5,31,1]
Properties	Vertex-transitive

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.

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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	↔
Cells	t{5,3} tr{5,3} t{3,3}
Faces	Triangle {3} square {4} hexagon {6} dodecagon {10}
Vertex figure	60px mirrored sphenoid
Coxeter group	DH3, [5,31,1]
Properties	Vertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h_2,3{4,3,5} and a mirrored sphenoid vertex figure.

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See also

• Convex uniform honeycombs in hyperbolic 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