# 5-cell honeycomb

4-simplex honeycomb :
| (No image) ||
| Type | Uniform 4-honeycomb |
| Family | Simplectic honeycomb |
| Schläfli symbol | {3$^{[5]}$} = 0$_{[5]}$ |
| Coxeter diagram | |
| 4-face types | {3,3,3} t$_1${3,3,3} |
| Cell types | {3,3} t$_1${3,3} |
| Face types | {3} |
| Vertex figure | t$_{0,3}${3,3,3} |
| Symmetry | ${\tilde {A}}_{4}$×2 [3$^{[5]}$] |
| Properties | vertex-transitive |
In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1.

## Structure

Cells of the vertex figure are ten tetrahedrons and 20 triangular prisms, corresponding to the ten 5-cells and 20 rectified 5-cells that meet at each vertex. All the vertices lie in parallel realms in which they form alternated cubic honeycombs, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell.

## Alternate names

* Cyclopentachoric tetracomb
* Pentachoric-dispentachoric tetracomb

## Projection by folding

The 5-cell honeycomb can be projected into the 2-dimensional square tiling by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
* ${\tilde {A}}_{3}$
* ${\tilde {C}}_{2}$
Two different aperiodic tilings with 5-fold symmetry can be obtained by projecting two-dimensional slices of the honeycomb: the Penrose tiling composed of rhombi, and the Tübingen triangle tiling composed of isosceles triangles.

## A4 lattice

The vertex arrangement of the 5-cell honeycomb is called the A4 lattice, or 4-simplex lattice. The 20 vertices of its vertex figure, the runcinated 5-cell represent the 20 roots of the ${\tilde {A}}_{4}$ Coxeter group. It is the 4-dimensional case of a simplectic honeycomb.
The A$^*$ $_4$ lattice is the union of five A$_4$ lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell
    ∪ ∪ ∪ ∪ = dual of

## Related polytopes and honeycombs

The tops of the 5-cells in this honeycomb adjoin the bases of the 5-cells, and vice versa, in adjacent laminae (or layers); but alternating laminae may be inverted so that the tops of the rectified 5-cells adjoin the tops of the rectified 5-cells and the bases of the 5-cells adjoin the bases of other 5-cells. This inversion results in another non-Wythoffian uniform convex honeycomb. Octahedral prisms and tetrahedral prisms may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.
This honeycomb is one of seven unique uniform honeycombs constructed by the ${\tilde {A}}_{4}$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:
A4 honeycombs :
| Pentagon symmetry | Extended symmetry | Extended group | Honeycomb diagrams |
| a1 | [3$^{[5]}$] | ${\tilde {A}}_{4}$ | (None) |
| i2 | [[3$^{[5]}$]] | ${\tilde {A}}_{4}$×2 | $_1$, $_2$, $_3$,  $_4$, $_5$, $_6$ |
| r10 | [5[3$^{[5]}$]] | ${\tilde {A}}_{4}$×10 | $_7$ |

### Rectified 5-cell honeycomb

Rectified 5-cell honeycomb :
| (No image) ||
| Type | Uniform 4-honeycomb |
| Schläfli symbol | t$_{0,2}${3$^{[5]}$} or r{3$^{[5]}$} |
| Coxeter diagram | |
| 4-face types | t$_1${3$^3$} t$_{0,2}${3$^3$} t$_{0,3}${3$^3$} |
| Cell types | Tetrahedron Octahedron Cuboctahedron Triangular prism |
| Vertex figure | triangular elongated-antiprismatic prism |
| Symmetry | ${\tilde {A}}_{4}$×2 [3$^{[5]}$] |
| Properties | vertex-transitive |
The rectified 4-simplex honeycomb or rectified 5-cell honeycomb is a space-filling tessellation honeycomb.

#### Alternate names

* small cyclorhombated pentachoric tetracomb
* small prismatodispentachoric tetracomb

### Cyclotruncated 5-cell honeycomb

Cyclotruncated 5-cell honeycomb :
| (No image) ||
| Type | Uniform 4-honeycomb |
| Family | Truncated simplectic honeycomb |
| Schläfli symbol | t$_{0,1}${3$^{[5]}$} |
| Coxeter diagram | |
| 4-face types | {3,3,3} t{3,3,3} 2t{3,3,3} |
| Cell types | {3,3} t{3,3} |
| Face types | Triangle {3} Hexagon {6} |
| Vertex figure | Tetrahedral antiprism [3,4,2$^+$], order 48 |
| Symmetry | ${\tilde {A}}_{4}$×2 [3$^{[5]}$] |
| Properties | vertex-transitive |
The cyclotruncated 4-simplex honeycomb or cyclotruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be seen as a birectified 5-cell honeycomb.
It is composed of 5-cells, truncated 5-cells, and bitruncated 5-cells facets in a ratio of 2:2:1. Its vertex figure is a tetrahedral antiprism, with 2 regular tetrahedron, 8 triangular pyramid, and 6 tetragonal disphenoid cells, defining 2 5-cell, 8 truncated 5-cell, and 6 bitruncated 5-cell facets around a vertex.
It can be constructed as five sets of parallel hyperplanes that divide space into two half-spaces. The 3-space hyperplanes contain quarter cubic honeycombs as a collection facets.

#### Alternate names

* Cyclotruncated pentachoric tetracomb
* Small truncated-pentachoric tetracomb

### Truncated 5-cell honeycomb

Truncated 4-simplex honeycomb :
| (No image) ||
| Type | Uniform 4-honeycomb |
| Schläfli symbol | t$_{0,1,2}${3$^{[5]}$} or t{3$^{[5]}$} |
| Coxeter diagram | |
| 4-face types | t$_{0,1}${3$^3$} t$_{0,1,2}${3$^3$} t$_{0,3}${3$^3$} |
| Cell types | Tetrahedron Truncated tetrahedron Truncated octahedron Triangular prism |
| Vertex figure | triangular elongated-antiprismatic pyramid |
| Symmetry | ${\tilde {A}}_{4}$×2 [3$^{[5]}$] |
| Properties | vertex-transitive |
The truncated 4-simplex honeycomb or truncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cyclocantitruncated 5-cell honeycomb.

#### Alaternate names

* Great cyclorhombated pentachoric tetracomb
* Great truncated-pentachoric tetracomb

### Cantellated 5-cell honeycomb

Cantellated 5-cell honeycomb :
| (No image) ||
| Type | Uniform 4-honeycomb |
| Schläfli symbol | t$_{0,1,3}${3$^{[5]}$} or rr{3$^{[5]}$} |
| Coxeter diagram | |
| 4-face types | t$_{0,2}${3$^3$} t$_{1,2}${3$^3$} t$_{0,1,3}${3$^3$} |
| Cell types | Truncated tetrahedron Octahedron Cuboctahedron Triangular prism Hexagonal prism |
| Vertex figure | Bidiminished rectified pentachoron |
| Symmetry | ${\tilde {A}}_{4}$×2 [3$^{[5]}$] |
| Properties | vertex-transitive |
The cantellated 4-simplex honeycomb or cantellated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cycloruncitruncated 5-cell honeycomb.

#### Alternate names

* Cycloprismatorhombated pentachoric tetracomb
* Great prismatodispentachoric tetracomb

### Bitruncated 5-cell honeycomb

Bitruncated 5-cell honeycomb :
| (No image) ||
| Type | Uniform 4-honeycomb |
| Schläfli symbol | t$_{0,1,2,3}${3$^{[5]}$} or 2t{3$^{[5]}$} |
| Coxeter diagram | |
| 4-face types | t$_{0,1,3}${3$^3$} t$_{0,1,2}${3$^3$} t$_{0,1,2,3}${3$^3$} |
| Cell types | Cuboctahedron Truncated octahedron Truncated tetrahedron Hexagonal prism Triangular prism |
| Vertex figure | tilted rectangular duopyramid |
| Symmetry | ${\tilde {A}}_{4}$×2 [3$^{[5]}$] |
| Properties | vertex-transitive |
The bitruncated 4-simplex honeycomb or bitruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be called a cycloruncicantitruncated 5-cell honeycomb.

#### Alternate names

* Great cycloprismated pentachoric tetracomb
* Grand prismatodispentachoric tetracomb

### Omnitruncated 5-cell honeycomb

Omnitruncated 4-simplex honeycomb :
| (No image) ||
| Type | Uniform 4-honeycomb |
| Family | Omnitruncated simplectic honeycomb |
| Schläfli symbol | t$_{0,1,2,3,4}${3$^{[5]}$} or tr{3$^{[5]}$} |
| Coxeter diagram | |
| 4-face types | t$_{0,1,2,3}${3,3,3} |
| Cell types | t$_{0,1,2}${3,3} {6}x |
| Face types | {4} {6} |
| Vertex figure | Irr. 5-cell |
| Symmetry | ${\tilde {A}}_{4}$×10, [5[3$^{[5]}$]] |
| Properties | vertex-transitive, cell-transitive |
The omnitruncated 4-simplex honeycomb or omnitruncated 5-cell honeycomb is a space-filling tessellation honeycomb. It can also be seen as a cyclosteriruncicantitruncated 5-cell honeycomb. .
It is composed entirely of omnitruncated 5-cell (omnitruncated 4-simplex) facets.
Coxeter calls this Hinton's honeycomb after C. H. Hinton, who described it in his book The Fourth Dimension in 1906.
The facets of all omnitruncated simplectic honeycombs are called permutohedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

#### Alternate names

* Omnitruncated cyclopentachoric tetracomb
* Great-prismatodecachoric tetracomb

#### A$_4$$^*$ lattice

The A$^*$ $_4$ lattice is the union of five A$_4$ lattices, and is the dual to the omnitruncated 5-cell honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell.
    ∪ ∪ ∪ ∪ = dual of

## Alternated form

This honeycomb can be alternated, creating omnisnub 5-cells with irregular 5-cells created at the deleted vertices. Although it is not uniform, the 5-cells have a symmetry of order 10.