8-demicubic honeycomb

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb.

It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h{4,3,3,3,3,3,3} and the alternated vertices create 8-orthoplex {3,3,3,3,3,3,4} facets.

D8 lattice
The vertex arrangement of the 8-demicubic honeycomb is the D8 lattice. The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice. The best known is 240, from the E8 lattice and the 521 honeycomb.

$${\tilde{E}}_8$$ contains $${\tilde{D}}_8$$ as a subgroup of index 270. Both $${\tilde{E}}_8$$ and $${\tilde{D}}_8$$ can be seen as affine extensions of $$D_8$$ from different nodes:

The D$$ lattice (also called D$$) can be constructed by the union of two D8 lattices. This packing is only a lattice for even dimensions. The kissing number is 240. (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8). It is identical to the E8 lattice. At 8-dimensions, the 240 contacts contain both the 27=128 from lower dimension contact progression (2n-1), and 16*7=112 from higher dimensions (2n(n-1)).

The D$$ lattice (also called D$$ and C$$) can be constructed by the union of all four D8 lattices: It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

The kissing number of the D$$ lattice is 16 (2n for n≥5). and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb,, containing all trirectified 8-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 256 8-demicube facets around each vertex.