7-demicubic honeycomb

The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb.

It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h{4,3,3,3,3,3} and the alternated vertices create 7-orthoplex {3,3,3,3,3,4} facets.

D7 lattice
The vertex arrangement of the 7-demicubic honeycomb is the D7 lattice. The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice. The best known is 126, from the E7 lattice and the 331 honeycomb.

The D$$ packing (also called D$$) can be constructed by the union of two D7 lattices. The D$$ packings form lattices only in even dimensions. The kissing number is 26=64 (2n-1 for n&lt;8, 240 for n=8, and 2n(n-1) for n&gt;8).

The D$$ lattice (also called D$$ and C$$) can be constructed by the union of all four 7-demicubic 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 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb,, containing all with tritruncated 7-orthoplex, Voronoi cells.

Symmetry constructions
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 128 7-demicube facets around each vertex.