Order-5 cubic honeycomb

Order-5 cubic honeycomb

Poincaré disk models
TypeHyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {4,3,5}
Coxeter diagram
Cells{4,3}
Facessquare {4}
Edge figurepentagon {5}
Vertex figure
icosahedron
Coxeter groupBH3, [5,3,4]
DualOrder-4 dodecahedral honeycomb
PropertiesRegular

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}

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}
{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:

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.

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 groupBH3, [5,3,4]
DH3, [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.

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
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}
square {4}
pentagon {5}
Vertex figure
pentagonal pyramid
Coxeter groupBH3, [5,3,4]
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}, with octahedral cells at the truncated vertices.

Related honeycombs

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

Same as 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 groupBH3, [5,3,4]
PropertiesVertex-transitive

The cantellated order-5 cubic honeycomb, , has rhombicuboctahedron and icosidodecahedron 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
TypeUniform honeycombs in hyperbolic space
Schläfli symboltr{4,3,5}
Coxeter diagram
Cellstr{4,3}
t{3,5}
Facessquare {4}
pentagon {5}
hexagon {6}
octahedron {8}
Vertex figure
Mirrored sphenoid
Coxeter groupBH3, [5,3,4]
PropertiesVertex-transitive

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

Related honeycombs

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
octahedron
Coxeter groupBH3, [5,3,4]
PropertiesVertex-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.

It is analogous 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}:

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
quad-pyramid
Coxeter groupBH3, [5,3,4]
PropertiesVertex-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.

Related honeycombs

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

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
tetrahedron
Coxeter groupBH3, [5,3,4]
PropertiesVertex-transitive

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

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
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh{4,3,5}
Coxeter diagram ↔
Cells{3,3}
{3,5}
Facestriangle {3}
pentagon {5}
Vertex figure
icosidodecahedron
Coxeter groupDH3, [5,31,1]
Propertiesquasiregular

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
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 groupDH3, [5,31,1]
PropertiesVertex-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.

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 prism
Coxeter groupDH3, [5,31,1]
PropertiesVertex-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.

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}
dodecagon {10}
Vertex figure
mirrored sphenoid
Coxeter groupDH3, [5,31,1]
PropertiesVertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h2,3{4,3,5} and a mirrored sphenoid vertex figure.

See also

References

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