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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 | Template:CDD or Template:CDD Template:CDD Template:CDD |
Facets | {3,3,3,3,3,4} h{4,3,3,3,3,3} |
Vertex figure | Rectified heptacross |
Coxeter group | [4,3,3,3,3,31,1] , [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 DTemplate:Sup sub packing (also called DTemplate:Sup sub) can be constructed by the union of two D7 lattices. The DTemplate:Sup sub 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 DTemplate:Sup sub lattice (also called DTemplate:Sup sub and CTemplate:Sup sub) 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 DTemplate:Sup sub lattice is 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb, Template:CDD, containing all with tritruncated 7-orthoplex, Template:CDD Voronoi cells.[5]
See also
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 [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D7.html
- ↑ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [2]
- ↑ Conway (1998), p. 119
- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds7.html
- ↑ Conway (1998), p. 466
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current | 20:11, 7 August 2007 | 868 × 538 (17 KB) | wikimediacommons>Bromskloss | {{Information |Description=A tree describing the process of throwing four Fudge dice and computing their sum. The uppermost row is the starting position (<math>0</math> dice thrown), the next row contains the possible |
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