7-demicubic honeycomb
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.
7-demicubic honeycomb | |
---|---|
(No image) | |
Type | Uniform 7-honeycomb |
Family | Alternated hypercube honeycomb |
Schläfli symbol | h{4,3,3,3,3,3,4} h{4,3,3,3,3,3^{1,1}} ht_{0,7}{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 | [4,3,3,3,3,3^{1,1}] , [3^{1,1},3,3,3,3^{1,1}] |
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 D_{7} 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 E_{7} lattice and the 3_{31} honeycomb.
The D^{+}
_{7} packing (also called D^{2}
_{7}) can be constructed by the union of two D_{7} lattices. The D^{+}
_{n} packings form lattices only in even dimensions. The kissing number is 2^{6}=64 (2^{n-1} for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]
∪
The D^{*}
_{7} lattice (also called D^{4}
_{7} and C^{2}
_{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,
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 |
---|---|---|---|---|
= [3^{1,1},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 | |
= [3^{1,1},3,3,3^{1,1}] = [1^{+},4,3,3,3,3^{1,1}] | h{4,3,3,3,3,3^{1,1}} | [3^{5,1,1}] |
64+64: 7-demicube 14: 7-orthoplex | |
2×½ = [[(4,3,3,3,3,4,2<sup>+</sup>)]] | ht_{0,7}{4,3,3,3,3,3,4} | 64+32+32: 7-demicube 14: 7-orthoplex |
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}={3^{1,1},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
- (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
- 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
- Conway (1998), p. 119
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds7.html
- Conway (1998), p. 466
External links
Fundamental convex regular and uniform honeycombs in dimensions 2-9 | ||||||
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Space | Family | / / | ||||
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |