# 5-cell honeycomb

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.

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.^{[1]}

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:

The A^{*}_{4} lattice^{[4]} 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

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.^{[5]}

The **rectified 4-simplex honeycomb** or **rectified 5-cell honeycomb** is a space-filling tessellation honeycomb.

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 an Elongated tetrahedral antiprism, with 8 equilateral triangle and 24 isosceles triangle faces, defining 8 5-cell and 24 truncated 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.^{[7]}

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**.

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**.

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**.

The **omnitruncated 4-simplex honeycomb** or **omnitruncated 5-cell honeycomb** is a space-filling tessellation honeycomb. It can also be seen as a **cantitruncated 5-cell honeycomb** and also 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.^{[8]}

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).

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.^{[9]}

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.