6-demicubic honeycomb
# 6-demicubic honeycomb

6-demicubic honeycomb :

| (No image) ||
| - | - |
| Type | Uniform 6-honeycomb |
| Family | Alternated hypercube honeycomb |
| Schläfli symbol | h{4,3,3,3,3,4} $h{4,3,3,3,3^{1,1}}$ $ht_{0,6}{4,3,3,3,3,4}$ |
| Coxeter diagram | = = |
| Facets | {3,3,3,3,4} h{4,3,3,3,3} |
| Vertex figure | r{3,3,3,3,4} |
| Coxeter group | ${\tilde {B}}_{6}$ $[4,3,3,3,3^{1,1}]$ ${\tilde {D}}_{6}$ $[3^{1,1},3,3,3^{1,1}]$ |

The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.

It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h{4,3,3,3,3} and the alternated vertices create 6-orthoplex {3,3,3,3,4} facets.

## D6 lattice

The vertex arrangement of the 6-demicubic honeycomb is the $D_6$ lattice. The 60 vertices of the rectified 6-orthoplex vertex figure of the 6-demicubic honeycomb reflect the kissing number 60 of this lattice. The best known is 72, from the $E_6$ lattice and the $2_{22}$ honeycomb.

The $D^+$ $_6$ lattice (also called $D^2$ $_6)$ can be constructed by the union of two $D_6$ lattices. This packing is only a lattice for even dimensions. The kissing number is $2^5=32$ $(2^{n-1}$ for n<8, 240 for n=8, and 2n(n-1) for n>8).

The $D^*$ $_6$ lattice (also called $D^4$ $_6$ and $C^2$ $_6)$ can be constructed by the union of all four 6-demicubic lattices: It is also the 6-dimensional body centered cubic, the union of two 6-cube honeycombs in dual positions.

The kissing number of the $D_6^*$ lattice is 12 (2n for n≥5). and its Voronoi tessellation is a trirectified 6-cubic honeycomb, , containing all birectified 6-orthoplex Voronoi cell, .

## Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 64 6-demicube facets around each vertex.

| Coxeter group | Schläfli symbol | Coxeter-Dynkin diagram | Vertex figure Symmetry | Facets/verf |
| - | - | - | - | - |
| ${\tilde {B}}_{6}$ = $[3^{1,1},3,3,3,4]$ = $[1^+,4,3,3,3,3,4]$ | h{4,3,3,3,3,4} | = | [3,3,3,4] | 64: 6-demicube 12: 6-orthoplex |
| ${\tilde {D}}_{6}$ = $[3^{1,1},3,3^{1,1}]$ = $[1^+,4,3,3,3^{1,1}]$ | $h{4,3,3,3,3^{1,1}}$ | = | $[3^{3,1,1}]$ | 32+32: 6-demicube 12: 6-orthoplex |
| ½${\tilde {C}}_{6}$ = $[[(4,3,3,3,4,2^+)]]$ | $ht_{0,6}{4,3,3,3,3,4}$ | | | 32+16+16: 6-demicube 12: 6-orthoplex |

## Related honeycombs

This honeycomb is one of 41 uniform honeycombs constructed by the ${\tilde {D}}_{6}$ Coxeter group, all but 6 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 41 permutations are listed with its highest extended symmetry, and related ${\tilde {B}}_{6}$ and ${\tilde {C}}_{6}$ constructions:

D6 honeycombs :

| Extended symmetry | Extended diagram | Order | Honeycombs |
| - | - | - | - |
| $[3^{1,1},3,3,3^{1,1}]$ | | ×1 | , |
| $[[3^{1,1},3,3,3^{1,1}]]$ | | ×2 | , , , |
| $<[3^{1,1},3,3,3^{1,1}]>$ ↔ $[3^{1,1},3,3,3,4]$ | ↔ | ×2 | , , , , , , , , , , , , , , , |
| $<2[3^{1,1},3,3,3^{1,1}]>$ ↔ [4,3,3,3,3,4] | ↔ | ×4 | ,, ,, , , , , , , , |
| $[<2[3^{1,1},3,3,3^{1,1}]>]$ ↔ [[4,3,3,3,3,4]] | ↔ | ×8 | , , , , , , |
