Order-5 cubic honeycomb

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.

Order-5 cubic honeycomb

Poincaré disk models
TypeHyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol{4,3,5}
Coxeter diagram
Cells{4,3} (cube)
Faces{4} (square)
Edge figure{5} (pentagon)
Vertex figure
icosahedron
Coxeter groupBH3, [4,3,5]
DualOrder-4 dodecahedral honeycomb
PropertiesRegular

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}

One cell, centered in Poincare ball model

Main cells

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.

The 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:

Four regular compact honeycombs in H3

{5,3,4}

{4,3,5}

{3,5,3}

{5,3,5}

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the 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}
{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}

The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral 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}
{8,3,5}
... {,3,5}
Image
Cells
{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{,3}

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.

{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}


{3,5}

{3,6}


{3,7}

{3,8}


{3,}

Rectified order-5 cubic honeycomb

Rectified order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolr{4,3,5} or 2r{5,3,4}
2r{5,31,1}
Coxeter diagram
Cellsr{4,3}
{3,5}
Facestriangle {3}
square {4}
Vertex figure
pentagonal prism
Coxeter group, [4,3,5]
, [5,31,1]
PropertiesVertex-transitive, edge-transitive

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

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
Symbols r{5,3,4}
r{4,3,5}
r{3,5,3}
r{5,3,5}
Vertex
figure
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
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
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt{4,3,5}
Coxeter diagram
Cellst{4,3}
{3,5}
Facestriangle {3}
octagon {8}
Vertex figure
pentagonal pyramid
Coxeter group, [4,3,5]
PropertiesVertex-transitive

The truncated 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 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}, which has octahedral cells at the truncated vertices.

Four truncated regular compact honeycombs in H3
Image
Symbols t{5,3,4}
t{4,3,5}
t{3,5,3}
t{5,3,5}
Vertex
figure

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
TypeUniform honeycombs in hyperbolic space
Schläfli symbolrr{4,3,5}
Coxeter diagram
Cellsrr{4,3}
r{3,5}
{}x{5}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter group, [4,3,5]
PropertiesVertex-transitive

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

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

Four cantellated regular compact honeycombs in H3
Image
Symbols rr{5,3,4}
rr{4,3,5}
rr{3,5,3}
rr{5,3,5}
Vertex
figure

Cantitruncated order-5 cubic honeycomb

Cantitruncated order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symboltr{4,3,5}
Coxeter diagram
Cellstr{4,3}
t{3,5}
{}x{5}
Facessquare {4}
pentagon {5}
hexagon {6}
octagon {8}
Vertex figure
mirrored sphenoid
Coxeter group, [4,3,5]
PropertiesVertex-transitive

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

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

Four cantitruncated regular compact honeycombs in H3
Image
Symbols tr{5,3,4}
tr{4,3,5}
tr{3,5,3}
tr{5,3,5}
Vertex
figure

Runcinated order-5 cubic honeycomb

Runcinated order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbolt0,3{4,3,5}
Coxeter diagram
Cells{4,3}
{5,3}
{}x{5}
Facessquare {4}
pentagon {5}
Vertex figure
irregular triangular antiprism
Coxeter group, [4,3,5]
PropertiesVertex-transitive

The runcinated order-5 cubic honeycomb or runcinated 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:

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

Three runcinated regular compact honeycombs in H3
Image
Symbols t0,3{4,3,5}
t0,3{3,5,3}
t0,3{5,3,5}
Vertex
figure

Runcitruncated order-5 cubic honeycomb

Runctruncated order-5 cubic honeycomb
Runcicantellated order-4 dodecahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,3{4,3,5}
Coxeter diagram
Cellst{4,3}
rr{5,3}
{}x{5}
{}x{8}
Facestriangle {3}
square {4}
pentagon {5}
octagon {8}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter group, [4,3,5]
PropertiesVertex-transitive

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-trapezoidal pyramid vertex figure.

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

Four runcitruncated regular compact honeycombs in H3
Image
Symbols t0,1,3{5,3,4}
t0,1,3{4,3,5}
t0,1,3{3,5,3}
t0,1,3{5,3,5}
Vertex
figure

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
TypeUniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbolt0,1,2,3{4,3,5}
Coxeter diagram
Cellstr{5,3}
tr{4,3}
{10}x{}
{8}x{}
Facessquare {4}
hexagon {6}
octagon {8}
decagon {10}
Vertex figure
irregular tetrahedron
Coxeter group, [4,3,5]
PropertiesVertex-transitive

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 tetrahedral vertex figure.

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

Three omnitruncated regular compact honeycombs in H3
Image
Symbols t0,1,2,3{4,3,5}
t0,1,2,3{3,5,3}
t0,1,2,3{5,3,5}
Vertex
figure

Alternated order-5 cubic honeycomb

Alternated order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh{4,3,5}
Coxeter diagram
Cells{3,3}
{3,5}
Facestriangle {3}
Vertex figure
icosidodecahedron
Coxeter group, [5,31,1]
PropertiesVertex-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.

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
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh2{4,3,5}
Coxeter diagram
Cellsr{5,3}
t{3,5}
t{3,3}
Facestriangle {3}
pentagon {5}
hexagon {6}
Vertex figure
rectangular pyramid
Coxeter group, [5,31,1]
PropertiesVertex-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.

Runcic order-5 cubic honeycomb

Runcic order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh3{4,3,5}
Coxeter diagram
Cells{5,3}
rr{5,3}
{3,3}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
triangular frustum
Coxeter group, [5,31,1]
PropertiesVertex-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.

Runcicantic order-5 cubic honeycomb

Runcicantic order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh2,3{4,3,5}
Coxeter diagram
Cellst{5,3}
tr{5,3}
t{3,3}
Facestriangle {3}
square {4}
hexagon {6}
decagon {10}
Vertex figure
irregular tetrahedron
Coxeter group, [5,31,1]
PropertiesVertex-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.

See also

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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