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Order-5 cubic honeycomb

March 28, 2023

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}

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.

Related polytopes and honeycombs

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:

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.

{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
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
pentagonal prism
Coxeter group , [4,3,5]
, [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.

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:

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
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{4,3,5}
Coxeter diagram
Cells t{4,3}
{3,5}
Faces triangle {3}
octagon {8}
Vertex figure
pentagonal pyramid
Coxeter group , [4,3,5]
Properties Vertex-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.

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

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

Related honeycombs

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

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}
{}x{5}
Faces square {4}
pentagon {5}
hexagon {6}
octagon {8}
Vertex figure
mirrored sphenoid
Coxeter group , [4,3,5]
Properties Vertex-transitive

The cantitruncated 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 (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:

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
irregular triangular antiprism
Coxeter group , [4,3,5]
Properties Vertex-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:

Related honeycombs

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

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
isosceles-trapezoidal pyramid
Coxeter group , [4,3,5]
Properties Vertex-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.

Related honeycombs

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

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
Cells tr{5,3}
tr{4,3}
{10}x{}
{8}x{}
Faces square {4}
hexagon {6}
octagon {8}
decagon {10}
Vertex figure
irregular tetrahedron
Coxeter group , [4,3,5]
Properties Vertex-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.

Related honeycombs

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

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}
Vertex figure
icosidodecahedron
Coxeter group , [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.

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
rectangular pyramid
Coxeter group , [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.

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
triangular frustum
Coxeter group , [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.

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}
decagon {10}
Vertex figure
irregular tetrahedron
Coxeter group , [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.

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

March 28, 2023
order, cubic, honeycomb, regular, tiling, hyperbolic, space, poincaré, disk, models, type, hyperbolic, regular, honeycombuniform, hyperbolic, honeycomb, schläfli, symbol, coxeter, diagram, cells, cube, faces, square, edge, figure, pentagon, vertex, figure, ico. Regular tiling of hyperbolic 3 space Order 5 cubic honeycomb Poincare disk models Type Hyperbolic regular honeycombUniform hyperbolic honeycomb Schlafli symbol 4 3 5 Coxeter diagram Cells 4 3 cube Faces 4 square Edge figure 5 pentagon Vertex figure icosahedron Coxeter group BH 3 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 Schlafli 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 Contents 1 Description 2 Symmetry 3 Related polytopes and honeycombs 3 1 Rectified order 5 cubic honeycomb 3 1 1 Related honeycomb 3 2 Truncated order 5 cubic honeycomb 3 2 1 Related honeycombs 3 3 Bitruncated order 5 cubic honeycomb 3 4 Cantellated order 5 cubic honeycomb 3 4 1 Related honeycombs 3 5 Cantitruncated order 5 cubic honeycomb 3 5 1 Related honeycombs 3 6 Runcinated order 5 cubic honeycomb 3 6 1 Related honeycombs 3 7 Runcitruncated order 5 cubic honeycomb 3 7 1 Related honeycombs 3 8 Runcicantellated order 5 cubic honeycomb 3 9 Omnitruncated order 5 cubic honeycomb 3 9 1 Related honeycombs 3 10 Alternated order 5 cubic honeycomb 3 10 1 Related honeycombs 3 11 Cantic order 5 cubic honeycomb 3 12 Runcic order 5 cubic honeycomb 3 13 Runcicantic order 5 cubic honeycomb 4 See also 5 References Description Edit 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 Edit It has a radial subgroup symmetry construction with dodecahedral fundamental domains Coxeter notation 4 3 5 index 120 Related polytopes and honeycombs Edit 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 8734 3 5 Image Cells 3 3 4 3 5 3 6 3 7 3 8 3 8734 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 8734 Image Vertexfigure 3 3 3 4 3 5 3 6 3 7 3 8 3 8734 Rectified order 5 cubic honeycomb Edit Rectified order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schlafli 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 pentagonal prism Coxeter group B H x00AF 3 displaystyle overline BH 3 4 3 5 D H x00AF 3 displaystyle overline DH 3 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 Related honeycomb Edit 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 Vertexfigure 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 8734 3 5 Image Cells 3 5 r 3 3 r 4 3 r 5 3 r 6 3 r 7 3 r 8734 3 Truncated order 5 cubic honeycomb Edit Truncated order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schlafli symbol t 4 3 5 Coxeter diagram Cells t 4 3 3 5 Faces triangle 3 octagon 8 Vertex figure pentagonal pyramid Coxeter group B H x00AF 3 displaystyle overline BH 3 4 3 5 Properties Vertex 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 Related honeycombs Edit 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 Vertexfigure Bitruncated order 5 cubic honeycomb Edit The bitruncated order 5 cubic honeycomb is the same as the bitruncated order 4 dodecahedral honeycomb Cantellated order 5 cubic honeycomb Edit Cantellated order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schlafli 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 wedge Coxeter group B H x00AF 3 displaystyle overline BH 3 4 3 5 Properties Vertex transitive The cantellated order 5 cubic honeycomb has rhombicuboctahedron icosidodecahedron and pentagonal prism cells with a wedge vertex figure Related honeycombs Edit 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 Vertexfigure Cantitruncated order 5 cubic honeycomb Edit Cantitruncated order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schlafli symbol tr 4 3 5 Coxeter diagram Cells tr 4 3 t 3 5 x 5 Faces square 4 pentagon 5 hexagon 6 octagon 8 Vertex figure mirrored sphenoid Coxeter group B H x00AF 3 displaystyle overline BH 3 4 3 5 Properties Vertex transitive The cantitruncated order 5 cubic honeycomb has truncated cuboctahedron truncated icosahedron and pentagonal prism cells with a mirrored sphenoid vertex figure Related honeycombs Edit 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 Vertexfigure Runcinated order 5 cubic honeycomb Edit Runcinated order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic spaceSemiregular honeycomb Schlafli symbol t0 3 4 3 5 Coxeter diagram Cells 4 3 5 3 x 5 Faces square 4 pentagon 5 Vertex figure irregular triangular antiprism Coxeter group B H x00AF 3 displaystyle overline BH 3 4 3 5 Properties Vertex 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 Related honeycombs Edit 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 Vertexfigure Runcitruncated order 5 cubic honeycomb Edit Runctruncated order 5 cubic honeycombRuncicantellated order 4 dodecahedral honeycomb Type Uniform honeycombs in hyperbolic space Schlafli 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 isosceles trapezoidal pyramid Coxeter group B H x00AF 3 displaystyle overline BH 3 4 3 5 Properties Vertex 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 Related honeycombs Edit 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 Vertexfigure Runcicantellated order 5 cubic honeycomb Edit The runcicantellated order 5 cubic honeycomb is the same as the runcitruncated order 4 dodecahedral honeycomb Omnitruncated order 5 cubic honeycomb Edit Omnitruncated order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic spaceSemiregular honeycomb Schlafli 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 irregular tetrahedron Coxeter group B H x00AF 3 displaystyle overline BH 3 4 3 5 Properties Vertex 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 Related honeycombs Edit 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 Vertexfigure Alternated order 5 cubic honeycomb Edit Alternated order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schlafli symbol h 4 3 5 Coxeter diagram Cells 3 3 3 5 Faces triangle 3 Vertex figure icosidodecahedron Coxeter group D H x00AF 3 displaystyle overline DH 3 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 Schlafli 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 Edit 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 Edit Cantic order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schlafli 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 rectangular pyramid Coxeter group D H x00AF 3 displaystyle overline DH 3 5 31 1 Properties Vertex transitive The cantic order 5 cubic honeycomb is a uniform compact space filling tessellation or honeycomb with Schlafli 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 Edit Runcic order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schlafli symbol h3 4 3 5 Coxeter diagram Cells 5 3 rr 5 3 3 3 Faces triangle 3 square 4 pentagon 5 Vertex figure triangular frustum Coxeter group D H x00AF 3 displaystyle overline DH 3 5 31 1 Properties Vertex transitive The runcic order 5 cubic honeycomb is a uniform compact space filling tessellation or honeycomb with Schlafli symbol h3 4 3 5 It has dodecahedron rhombicosidodecahedron and tetrahedron cells with a triangular frustum vertex figure Runcicantic order 5 cubic honeycomb Edit Runcicantic order 5 cubic honeycomb Type Uniform honeycombs in hyperbolic space Schlafli 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 decagon 10 Vertex figure irregular tetrahedron Coxeter group D H x00AF 3 displaystyle overline DH 3 5 31 1 Properties Vertex transitive The runcicantic order 5 cubic honeycomb is a uniform compact space filling tessellation or honeycomb with Schlafli symbol h2 3 4 3 5 It has truncated dodecahedron truncated icosidodecahedron and truncated tetrahedron cells with an irregular tetrahedron vertex figure See also Edit Convex uniform honeycombs in hyperbolic space Regular tessellations of hyperbolic 3 space References Edit Coxeter Regular Polytopes 3rd ed Dover Publications 1973 ISBN 160 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 160 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 Retrieved from https en wikipedia org w index php title Order 5 cubic honeycomb amp oldid 1119308548 Alternated order 5 cubic honeycomb, wikipedia, wiki, book, books, library,

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