# Order-5 cubic honeycomb

Order-5 cubic honeycomb

Poincaré disk models
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {4,3,5}
Coxeter-Dynkin diagram
Cells cube {4,3}
Faces square {4}
Edge figure pentagon {5}
Vertex figure
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.

## Symmetry

It a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.

## Related polytopes and honeycombs

It is related to the regular (order-4) cubic honeycomb of Euclidean 3-space, which has 4 cubes per edge, and also the tesseract of Euclidean 4-space with 3 cubes per edge.

It has a related alternation honeycomb, represented by = , having icosahedron and tetrahedron cells.

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

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

{p,3,5}
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}

It in a sequence of regular polychora and honeycombs with cubic cells.

{4,3,p}
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-4 dodecahedral honeycomb

Rectified order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{4,3,5}
2r{5,31,1}
Coxeter diagram
=
Cells Cuboctahedron r{4,3}
Icosahedron {3,5}
Faces triangle {3}
square {4}
Vertex figure
pentagonal prism
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-uniform

The rectified order-4 dodecahedral honeycomb, , has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.