# Simplicial honeycomb

| ${\tilde {A}}_{2}$ | ${\tilde {A}}_{3}$ |
| Triangular tiling | Tetrahedral-octahedral honeycomb |
| With red and yellow equilateral triangles | With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra) |
In geometry, the simplicial honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the ${\tilde {A}}_{n}$ affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of n + 1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes $x+y+\cdots \in \mathbb {Z}$, then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.
In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph , with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph , filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.

## By dimension

| n | ${\tilde {A}}_{2+}$ | Tessellation | Vertex figure | Facets per vertex figure | Vertices per vertex figure | Edge figure |
| 1 | ${\tilde {A}}_{1}$ | Apeirogon | Line segment | 2 | 2 | Point |
| 2 | ${\tilde {A}}_{2}$ | Triangular tiling 2-simplex honeycomb | Hexagon (Truncated triangle) | 3+3 triangles | 6 | Line segment |
| 3 | ${\tilde {A}}_{3}$ | Tetrahedral-octahedral honeycomb 3-simplex honeycomb | Cuboctahedron (Cantellated tetrahedron) | 4+4 tetrahedron 6 rectified tetrahedra | 12 | Rectangle |
| 4 | ${\tilde {A}}_{4}$ | 4-simplex honeycomb | Runcinated 5-cell | 5+5 5-cells 10+10 rectified 5-cells | 20 | Triangular antiprism |
| 5 | ${\tilde {A}}_{5}$ | 5-simplex honeycomb | Stericated 5-simplex | 6+6 5-simplex 15+15 rectified 5-simplex 20 birectified 5-simplex | 30 | Tetrahedral antiprism |
| 6 | ${\tilde {A}}_{6}$ | 6-simplex honeycomb | Pentellated 6-simplex | 7+7 6-simplex 21+21 rectified 6-simplex 35+35 birectified 6-simplex | 42 | 4-simplex antiprism |
| 7 | ${\tilde {A}}_{7}$ | 7-simplex honeycomb | Hexicated 7-simplex | 8+8 7-simplex 28+28 rectified 7-simplex 56+56 birectified 7-simplex 70 trirectified 7-simplex | 56 | 5-simplex antiprism |
| 8 | ${\tilde {A}}_{8}$ | 8-simplex honeycomb | Heptellated 8-simplex | 9+9 8-simplex 36+36 rectified 8-simplex 84+84 birectified 8-simplex 126+126 trirectified 8-simplex | 72 | 6-simplex antiprism |
| 9 | ${\tilde {A}}_{9}$ | 9-simplex honeycomb | Octellated 9-simplex | 10+10 9-simplex 45+45 rectified 9-simplex 120+120 birectified 9-simplex 210+210 trirectified 9-simplex 252 quadrirectified 9-simplex | 90 | 7-simplex antiprism |
| 10 | ${\tilde {A}}_{10}$ | 10-simplex honeycomb | Ennecated 10-simplex | 11+11 10-simplex 55+55 rectified 10-simplex 165+165 birectified 10-simplex 330+330 trirectified 10-simplex 462+462 quadrirectified 10-simplex | 110 | 8-simplex antiprism |
| 11 | ${\tilde {A}}_{11}$ | 11-simplex honeycomb | ... | ... | ... | ... |

## Projection by folding

The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
| ${\tilde {A}}_{2}$ | ${\tilde {A}}_{4}$ | ${\tilde {A}}_{6}$ | ${\tilde {A}}_{8}$ | ${\tilde {A}}_{10}$ | ... |
| ${\tilde {A}}_{3}$ | ${\tilde {A}}_{3}$ | ${\tilde {A}}_{5}$ | ${\tilde {A}}_{7}$ | ${\tilde {A}}_{9}$ | ... |
| ${\tilde {C}}_{1}$ | ${\tilde {C}}_{2}$ | ${\tilde {C}}_{3}$ | ${\tilde {C}}_{4}$ | ${\tilde {C}}_{5}$ | ... |

## Kissing number

These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the vertex figure. This represents the highest kissing number for 2 and 3 dimensions, but falls short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.