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.
The vertex arrangement of the 7-demicubic honeycomb is the D<sub>7</sub> 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 E<sub>7</sub> lattice and the 3<sub>31</sub> honeycomb.
The D packing (also called D) can be constructed by the union of two D<sub>7</sub> lattices. The D packings form lattices only in even dimensions. The kissing number is 2<sup>6</sup>=64 (2<sup>n-1</sup> for n<8, 240 for n=8, and 2n(n-1) for n>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 nâÂÂ¥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb, , containing all with tritruncated 7-orthoplex, Voronoi cells.
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.