# 7-demicubic honeycomb

7-demicubic honeycomb
(No image)
Type Uniform 7-honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol h{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 figure Rectified 7-orthoplex
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 D27) 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 D47 and C27) 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 different 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
2×½$\displaystyle{ {\tilde{C}}_7 }$ = (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. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
2. Conway (1998), p. 119
3. Conway (1998), p. 466