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.
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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+
7 packing (also called D2
7) can be constructed by the union of two D7 lattices. The D+
n packings form lattices only in even dimensions. The kissing number is 26=64 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).
The D*
7 lattice (also called D4
7 and C2
7) 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*
7 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.