# Simplectic honeycomb

In geometry, the simplectic honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the ${\tilde {A}}_{n}$ affine Coxeter group symmetry. It is given a Schläfli symbol {3[n+1]}, and 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 $x+y+...\in \mathbb {Z}$ , 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.

${\tilde {A}}_{2}$ ${\tilde {A}}_{3}$ Triangular tiling Tetrahedral-octahedral honeycomb

With red and yellow equilateral triangles

With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra)

In 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
1 2 -
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
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
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:

${\tilde {A}}_{2}$ ... ... ...

## 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. For 2 and 3 dimensions, this represents the highest kissing number for 2 and 3 dimensions, but fall 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.