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7-demicubic honeycomb

Uniform 7-Honeycomb

7-demicubic honeycomb
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Type
Family
Schläfli symbol
Coxeter-Dynkin diagram
Facets
Vertex figure
Coxeter group

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. The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice.Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai https://books.google.com/books?id=upYwZ6cQumoC&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19 The best known is 126, from the E7 lattice and the 331 honeycomb.

The D packing (also called D) 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 (2n-1 for n8). : ∪

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 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 figureSymmetry
{\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-demicube14: 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-demicube14: 7-orthoplex
2×½{\tilde{C}}_7 = (4,3,3,3,3,4,2+)	ht0,7{4,3,3,3,3,3,4}			64+32+32: 7-demicube14: 7-orthoplex

References

Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
- 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Notes

References

"The Lattice D7".
Conway (1998), p. 119
"The Lattice D7".
Conway (1998), p. 466

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honeycombs-(geometry) 8-polytopes

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