# 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.

7-demicubic honeycomb
(No image)
TypeUniform 7-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbolh{4,3,3,3,3,3,4}
h{4,3,3,3,3,31,1}
ht0,7{4,3,3,3,3,3,4}
Coxeter-Dynkin diagram =
=
Facets{3,3,3,3,3,4}
h{4,3,3,3,3,3}
Vertex figureRectified 7-orthoplex
Coxeter group${\tilde {B}}_{7}$ [4,3,3,3,3,31,1]
${\tilde {D}}_{7}$ , [31,1,3,3,3,31,1]

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.

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
${\tilde {B}}_{7}$ = [31,1,3,3,3,3,4]
= [1+,4,3,3,3,3,3,4]
h{4,3,3,3,3,3,4} =
[3,3,3,3,3,4]
128: 7-demicube
14: 7-orthoplex
${\tilde {D}}_{7}$ = [31,1,3,3,31,1]
= [1+,4,3,3,3,31,1]
h{4,3,3,3,3,31,1} =
[35,1,1]
64+64: 7-demicube
14: 7-orthoplex
2×½${\tilde {C}}_{7}$ = [[(4,3,3,3,3,4,2<sup>+</sup>)]]ht0,7{4,3,3,3,3,3,4} 64+32+32: 7-demicube
14: 7-orthoplex

## See also

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