7-demicubic honeycomb


The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation 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 and the alternated vertices create 7-orthoplex 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 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.
The D lattice 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 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.
Coxeter groupSchläfli symbolCoxeter-Dynkin diagramVertex figure
Symmetry		Facets/verf
=
= 		h =
128: 7-demicube
14: 7-orthoplex
=
= 		h =
64+64: 7-demicube
14: 7-orthoplex
2×½ = ht0,764+32+32: 7-demicube
14: 7-orthoplex