| bgcolor=#e7dcc3 colspan=2 | 6-demicubic honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) | |
| Type | Uniform 6-honeycomb | |
| Family | Alternated hypercube honeycomb | |
| Schläfli symbol | h h ht0,6 | |
| Coxeter diagram | = = | |
| Facets | ||
| Vertex figure | ||
| Coxeter group | {\tilde{B}}6 {\tilde{D}}6 |
It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h and the alternated vertices create 6-orthoplex facets.
The vertex arrangement of the 6-demicubic honeycomb is the D6 lattice.[1] The 60 vertices of the rectified 6-orthoplex vertex figure of the 6-demicubic honeycomb reflect the kissing number 60 of this lattice.[2] The best known is 72, from the E6 lattice and the 222 honeycomb.
The D lattice (also called D) can be constructed by the union of two D6 lattices. This packing is only a lattice for even dimensions. The kissing number is 25=32 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]
∪
The D lattice (also called D and C) can be constructed by the union of all four 6-demicubic lattices:[4] It is also the 6-dimensional body centered cubic, the union of two 6-cube honeycombs in dual positions.
∪ ∪ ∪ = ∪ .
The kissing number of the D6* lattice is 12 (2n for n≥5).[5] and its Voronoi tessellation is a trirectified 6-cubic honeycomb,, containing all birectified 6-orthoplex Voronoi cell, .[6]
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 = [1<sup>+</sup>,4,3,3,3,3,4] | h | = | [3,3,3,4] | 64: 6-demicube 12: 6-orthoplex | |
{\tilde{D}}6 = [1<sup>+</sup>,4,3,3,3<sup>1,1</sup>] | h | = | [3<sup>3,1,1</sup>] | 32+32: 6-demicube 12: 6-orthoplex | |
| ½ {\tilde{C}}6 | ht0,6 | 32+16+16: 6-demicube 12: 6-orthoplex |