# 7-demicubic honeycomb

7-demicubic honeycomb
(No image)
Type Uniform honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol h{4,3,3,3,3,3,4}
Coxeter-Dynkin diagram or

Facets {3,3,3,3,3,4}
h{4,3,3,3,3,3}
Vertex figure Rectified heptacross
Coxeter group ${\displaystyle {\tilde {B}}_{7}}$ [4,3,3,3,3,31,1]
${\displaystyle {\tilde {D}}_{7}}$, [31,1,3,3,3,31,1]

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.[1] The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice.[2] 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).[3]

The D*
7
lattice (also called D4
7
and C2
7
) can be constructed by the union of all four 7-demicubic lattices:[4] 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.[5]

## Symmetry constructions

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

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
${\displaystyle {\tilde {B}}_{7}}$ = [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
${\displaystyle {\tilde {D}}_{7}}$ = [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
${\displaystyle {\tilde {C}}_{7}}$ = [[(4,3,3,3,3,4,2+)]] ht0,7{4,3,3,3,3,3,4} 64+32+32: 7-demicube
14: 7-orthoplex

## References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}, ...
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.

## Notes

1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D7.html
2. ^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
3. ^ Conway (1998), p. 119
4. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds7.html
5. ^ Conway (1998), p. 466