# Simplectic honeycomb

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

With red and yellow equilateral triangles

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

In geometry, the simplectic honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the ${\displaystyle {\tilde {A}}_{n}}$ 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 ${\displaystyle x+y+...\in \mathbb {Z} }$, 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.

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 ${\displaystyle {\tilde {A}}_{2+}}$  Tessellation Vertex figure Facets per vertex figure Vertices per vertex figure Edge figure
1 ${\displaystyle {\tilde {A}}_{1}}$
Apeirogon

1 2 -
2 ${\displaystyle {\tilde {A}}_{2}}$
Triangular tiling
2-simplex honeycomb

Hexagon
(Truncated triangle)

3+3 triangles 6 Line segment

3 ${\displaystyle {\tilde {A}}_{3}}$
Tetrahedral-octahedral honeycomb
3-simplex honeycomb

Cuboctahedron
(Cantellated tetrahedron)

4+4 tetrahedron
6 rectified tetrahedra
12
Rectangle

4 ${\displaystyle {\tilde {A}}_{4}}$  4-simplex honeycomb

Runcinated 5-cell

5+5 5-cells
10+10 rectified 5-cells
20
Triangular antiprism

5 ${\displaystyle {\tilde {A}}_{5}}$  5-simplex honeycomb

Stericated 5-simplex

6+6 5-simplex
15+15 rectified 5-simplex
20 birectified 5-simplex
30
Tetrahedral antiprism

6 ${\displaystyle {\tilde {A}}_{6}}$  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 ${\displaystyle {\tilde {A}}_{7}}$  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 ${\displaystyle {\tilde {A}}_{8}}$  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 ${\displaystyle {\tilde {A}}_{9}}$  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 ${\displaystyle {\tilde {A}}_{10}}$  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 ${\displaystyle {\tilde {A}}_{11}}$  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:

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

## 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.

## References

• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21