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7-demicubic honeycomb

Uniform 7-Honeycomb

Summary

Uniform 7-Honeycomb

7-demicubic honeycomb
(No image)
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

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

honeycombs-(geometry) 8-polytopes

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