6-demicubic honeycomb
The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.
6-demicubic honeycomb | |
---|---|
(No image) | |
Type | Uniform 6-honeycomb |
Family | Alternated hypercube honeycomb |
Schläfli symbol | h{4,3,3,3,3,4} h{4,3,3,3,3^{1,1}} ht_{0,6}{4,3,3,3,3,4} |
Coxeter diagram | |
Facets | {3,3,3,3,4} h{4,3,3,3,3} |
Vertex figure | r{3,3,3,3,4} |
Coxeter group | [4,3,3,3,3^{1,1}] [3^{1,1},3,3,3^{1,1}] |
It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h{4,3,3,3,3} and the alternated vertices create 6-orthoplex {3,3,3,3,4} facets.
D6 lattice
The vertex arrangement of the 6-demicubic honeycomb is the D_{6} lattice.[1] The 60 vertices of the rectified 6-orthoplex vertex figure of the 6-demicubic honeycomb reflect the kissing number 60 of this lattice.[2] The best known is 72, from the E_{6} lattice and the 2_{22} honeycomb.
The D^{+}
_{6} lattice (also called D^{2}
_{6}) can be constructed by the union of two D_{6} lattices. This packing is only a lattice for even dimensions. The kissing number is 2^{5}=32 (2^{n-1} for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]
∪
The D^{*}
_{6} lattice (also called D^{4}
_{6} and C^{2}
_{6}) can be constructed by the union of all four 6-demicubic lattices:[4] It is also the 6-dimensional body centered cubic, the union of two 6-cube honeycombs in dual positions.
∪ ∪ ∪ = ∪ .
The kissing number of the D_{6}^{*} lattice is 12 (2n for n≥5).[5] and its Voronoi tessellation is a trirectified 6-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 64 6-demicube facets around each vertex.
Coxeter group | Schläfli symbol | Coxeter-Dynkin diagram | Vertex figure Symmetry |
Facets/verf |
---|---|---|---|---|
= [3^{1,1},3,3,3,4] = [1^{+},4,3,3,3,3,4] | h{4,3,3,3,3,4} | [3,3,3,4] |
64: 6-demicube 12: 6-orthoplex | |
= [3^{1,1},3,3^{1,1}] = [1^{+},4,3,3,3^{1,1}] | h{4,3,3,3,3^{1,1}} | [3^{3,1,1}] |
32+32: 6-demicube 12: 6-orthoplex | |
½ = [[(4,3,3,3,4,2<sup>+</sup>)]] | ht_{0,6}{4,3,3,3,3,4} | 32+16+16: 6-demicube 12: 6-orthoplex |
Related honeycombs
This honeycomb is one of 41 uniform honeycombs constructed by the Coxeter group, all but 6 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 41 permutations are listed with its highest extended symmetry, and related and constructions:
D6 honeycombs | |||
---|---|---|---|
Extended symmetry |
Extended diagram |
Order | Honeycombs |
[3^{1,1},3,3,3^{1,1}] | ×1 | ||
[[3^{1,1},3,3,3^{1,1}]] | ×2 | ||
<[3^{1,1},3,3,3^{1,1}]> ↔ [3^{1,1},3,3,3,4] |
↔ |
×2 | |
<2[3^{1,1},3,3,3^{1,1}]> ↔ [4,3,3,3,3,4] |
↔ |
×4 | |
[<2[3^{1,1},3,3,3^{1,1}]>] ↔ [[4,3,3,3,3,4]] |
↔ |
×8 |
See also
Notes
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D6.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/Ds6.html
- Conway (1998), p. 120
- Conway (1998), p. 466
External links
- 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.
Fundamental convex regular and uniform honeycombs in dimensions 2-9 | ||||||
---|---|---|---|---|---|---|
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} |