Summary
| 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 |     =     =     
|
| 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
editThe 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 D+
7 packing (also called D2
7) can be constructed by the union of two D7 lattices. The D+
n 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 D*
7 lattice (also called D4
7 and C2
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, 
, containing all with tritruncated 7-orthoplex, 
Voronoi cells.[5]
Symmetry constructions
editThere 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} | = |
[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} | = |
[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} | |
64+32+32: 7-demicube 14: 7-orthoplex |
See also
editReferences
edit- 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 [2]
- (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
editExternal links
edit| Space | Family | / / | ||||
|---|---|---|---|---|---|---|
| E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
| E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
| En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |
