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 | Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD |
| Facets | {3,3,3,3,3,4} h{4,3,3,3,3,3} |
| Vertex figure | Rectified 7-orthoplex |
| Coxeter group | [4,3,3,3,3,31,1] , [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 DScript error: No such module "Su". packing (also called DScript error: No such module "Su".) can be constructed by the union of two D7 lattices. The DScript error: No such module "Su". 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 DScript error: No such module "Su". lattice (also called DScript error: No such module "Su". and CScript error: No such module "Su".) 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 DScript error: No such module "Su". 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]
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 |
|---|---|---|---|---|
| = [31,1,3,3,3,3,4] = [1+,4,3,3,3,3,3,4] |
h{4,3,3,3,3,3,4} | Template:CDD = Template:CDD | Template:CDD [3,3,3,3,3,4] |
128: 7-demicube 14: 7-orthoplex |
| = [31,1,3,3,31,1] = [1+,4,3,3,3,31,1] |
h{4,3,3,3,3,31,1} | Template:CDD = Template:CDD | Template:CDD [35,1,1] |
64+64: 7-demicube 14: 7-orthoplex |
| 2×½ = [[(4,3,3,3,3,4,2+)]] | ht0,7{4,3,3,3,3,3,4} | Template:CDD | 64+32+32: 7-demicube 14: 7-orthoplex |
See also
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Template:ISBN
- 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, Template:ISBN [2]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Script error: No such module "citation/CS1".
Notes
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