Order-5 cubic honeycomb

Order-5 cubic honeycomb
File:H3 435 CC center.png Poincaré disk models
Type	Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol	{4,3,5}
Coxeter diagram	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Cells	{4,3} (cube) File:Uniform polyhedron-43-t0.png
Faces	{4} (square)
Edge figure	{5} (pentagon)
Vertex figure	File:Order-5 cubic honeycomb verf.svg icosahedron
Coxeter group	BH3, [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 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

File:H2-5-4-primal.svg

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

File:Hyperb gcubic hc constr.png 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, File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png, 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

File:H3 534 CC center.png {5,3,4}

File:H3 435 CC center.png {4,3,5}

File:H3 353 CC center.png {3,5,3}

File:H3 535 CC center.png {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} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	r{5,3,4} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	t{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	rr{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	t0,3{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	tr{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	t0,1,3{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	t0,1,2,3{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png
File:H3 534 CC center.png	File:H3 534 CC center 0100.png	File:H3 435-0011 center ultrawide.png	File:H3 534-1010 center ultrawide.png	File:H3 534-1001 center ultrawide.png	File:H3 534-1110 center ultrawide.png	File:H3 534-1101 center ultrawide.png	File:H3 534-1111 center ultrawide.png
File:H3 435 CC center.png	File:H3 435 CC center 0100.png	File:H3 534-0011 center ultrawide.png	File:H3 534-0101 center ultrawide.png	File:H3 534-0110 center ultrawide.png	File:H3 534-0111 center ultrawide.png	File:H3 534-1011 center ultrawide.png
{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{4,3,5} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	t{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	rr{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png	2t{4,3,5} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png	tr{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png	t0,1,3{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png	t0,1,2,3{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.png

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

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.

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	File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node h0.png ↔ File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png
Cells	r{4,3} File:Uniform polyhedron-43-t1.png {3,5} File:Uniform polyhedron-53-t2.png
Faces	triangle {3} square {4}
Vertex figure	File:Rectified order-5 cubic honeycomb verf.png pentagonal prism
Coxeter group	${\overline{BH}}_3$, [4,3,5] ${\overline{DH}}_3$, [5,31,1]
Properties	Vertex-transitive, edge-transitive

The rectified order-5 cubic honeycomb, File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png, has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure. File:H3 435 CC center 0100.png

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:

Four rectified regular compact honeycombs in H³

Image	File:H3 534 CC center 0100.png	File:H3 435 CC center 0100.png	File:H3 353 CC center 0100.png	File:H3 535 CC center 0100.png
Symbols	r{5,3,4} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	r{4,3,5} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{3,5,3} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	r{5,3,5} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Vertex figure	File:Rectified order-4 dodecahedral honeycomb verf.png	File:Rectified order-5 cubic honeycomb verf.png	File:Rectified icosahedral honeycomb verf.png	File:Rectified order-5 dodecahedral honeycomb verf.png

r{p,3,5}

Space	S3	H3
Form	Finite	Compact		Paracompact	Noncompact
Name	r{3,3,5} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{4,3,5} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{5,3,5} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{6,3,5} File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	r{7,3,5} File:CDel node.pngFile:CDel 7.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	... r{∞,3,5} File:CDel node.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Image	File:Stereographic rectified 600-cell.png	File:H3 435 CC center 0100.png	File:H3 535 CC center 0100.png	File:H3 635 boundary 0100.png
Cells File:Icosahedron.png {3,5} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	File:Uniform polyhedron-33-t1.svg r{3,3} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Cuboctahedron.png r{4,3} File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Icosidodecahedron.png r{5,3} File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Uniform tiling 63-t1.svg r{6,3} File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Triheptagonal tiling.svg r{7,3} File:CDel node.pngFile:CDel 7.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:H2 tiling 23i-2.png r{∞,3} File:CDel node.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png

Truncated order-5 cubic honeycomb

Truncated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t{4,3,5}
Coxeter diagram	File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png
Cells	t{4,3} File:Uniform polyhedron-43-t01.png {3,5} File:Uniform polyhedron-53-t2.png
Faces	triangle {3} octagon {8}
Vertex figure	File:Truncated order-5 cubic honeycomb verf.png pentagonal pyramid
Coxeter group	${\overline{BH}}_3$, [4,3,5]
Properties	Vertex-transitive

The truncated order-5 cubic honeycomb, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png, 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.

File:Truncated cubic honeycomb.png

Related honeycombs

Four truncated regular compact honeycombs in H³

Image	File:H3 435-0011 center ultrawide.png	File:H3 534-0011 center ultrawide.png	File:H3 353-0011 center ultrawide.png	File:H3 535-0011 center ultrawide.png
Symbols	t{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	t{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	t{3,5,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	t{5,3,5} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Vertex figure	File:Truncated order-4 dodecahedral honeycomb verf.png	File:Truncated order-5 cubic honeycomb verf.png	File:Truncated icosahedral honeycomb verf.png	File:Truncated order-5 dodecahedral honeycomb verf.png

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	File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png
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	${\overline{BH}}_3$, [4,3,5]
Properties	Vertex-transitive

The cantellated order-5 cubic honeycomb, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png, 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

Cantitruncated order-5 cubic honeycomb

Cantitruncated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	tr{4,3,5}
Coxeter diagram	File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png
Cells	tr{4,3} File:Uniform polyhedron-43-t012.png 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	${\overline{BH}}_3$, [4,3,5]
Properties	Vertex-transitive

The cantitruncated order-5 cubic honeycomb, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png, 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

Four cantitruncated regular compact honeycombs in H³

Image	File:H3 534-1110 center ultrawide.png	File:H3 534-0111 center ultrawide.png	File:H3 353-1110 center ultrawide.png	File:H3 535-1110 center ultrawide.png
Symbols	tr{5,3,4} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	tr{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png	tr{3,5,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	tr{5,3,5} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png
Vertex figure	File:Cantitruncated order-4 dodecahedral honeycomb verf.png	File:Cantitruncated order-5 cubic honeycomb verf.png	File:Cantitruncated icosahedral honeycomb verf.png	File:Cantitruncated order-5 dodecahedral honeycomb verf.png

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	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png
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	File:Runcinated order-5 cubic honeycomb verf.png irregular triangular antiprism
Coxeter group	${\overline{BH}}_3$, [4,3,5]
Properties	Vertex-transitive

The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png, 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}, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png with square and pentagonal faces:

File:H2-5-4-cantellated.svg

Related honeycombs

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

File:Runcinated cubic honeycomb.png

Three runcinated regular compact honeycombs in H³

Image	File:H3 534-1001 center ultrawide.png	File:H3 353-1001 center ultrawide.png	File:H3 535-1001 center ultrawide.png
Symbols	t0,3{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png	t0,3{3,5,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	t0,3{5,3,5} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png
Vertex figure	File:Runcinated order-5 cubic honeycomb verf.png	File:Runcinated icosahedral honeycomb verf.png	File:Runcinated order-5 dodecahedral honeycomb verf.png

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	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png
Cells	t{4,3} File:Uniform polyhedron-43-t01.png rr{5,3} File:Uniform polyhedron-53-t02.png {}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	${\overline{BH}}_3$, [4,3,5]
Properties	Vertex-transitive

The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png, has truncated cube, rhombicosidodecahedron, pentagonal prism, and octagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure. File:H3 534-1011 center ultrawide.png

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

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	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png
Cells	tr{5,3} File:Uniform polyhedron-53-t012.png tr{4,3} File:Uniform polyhedron-43-t012.png {10}x{} File:Decagonal prism.png {8}x{} File:Octagonal prism.png
Faces	square {4} hexagon {6} octagon {8} decagon {10}
Vertex figure	File:Omnitruncated order-4 dodecahedral honeycomb verf.png irregular tetrahedron
Coxeter group	${\overline{BH}}_3$, [4,3,5]
Properties	Vertex-transitive

The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png, 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

Alternated order-5 cubic honeycomb

Alternated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h{4,3,5}
Coxeter diagram	File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Cells	{3,3} File:Uniform polyhedron-33-t0.png {3,5} File:Uniform polyhedron-53-t2.png
Faces	triangle {3}
Vertex figure	File:Alternated order-5 cubic honeycomb verf.png icosidodecahedron
Coxeter group	${\overline{DH}}_3$, [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 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. File:Alternated order 5 cubic honeycomb.png

Related honeycombs

It has 3 related forms: the cantic order-5 cubic honeycomb, File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png, the runcic order-5 cubic honeycomb, File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png, and the runcicantic order-5 cubic honeycomb, File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.png.

Cantic order-5 cubic honeycomb

Cantic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h2{4,3,5}
Coxeter diagram	File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png
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	${\overline{DH}}_3$, [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. File:H3 5311-0110 center ultrawide.png

Runcic order-5 cubic honeycomb

Runcic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h3{4,3,5}
Coxeter diagram	File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png
Cells	{5,3} File:Uniform polyhedron-53-t0.png rr{5,3} File:Uniform polyhedron-53-t02.png {3,3} File:Uniform polyhedron-33-t0.png
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	File:Runcinated alternated order-5 cubic honeycomb verf.png triangular frustum
Coxeter group	${\overline{DH}}_3$, [5,31,1]
Properties	Vertex-transitive

The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h₃{4,3,5}. It has dodecahedron, rhombicosidodecahedron, and tetrahedron cells, with a triangular frustum vertex figure. File:H3 5311-1010 center ultrawide.png

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	File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node 1.png
Cells	t{5,3} File:Uniform polyhedron-53-t01.png 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	File:Runcitruncated alternated order-5 cubic honeycomb verf.png irregular tetrahedron
Coxeter group	${\overline{DH}}_3$, [5,31,1]
Properties	Vertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with 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

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-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