Sums of powers
| 6-demicubic honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform honeycomb |
| Family | Alternated hypercube honeycomb |
| Schläfli symbol | h{4,3,3,3,3,4} |
| Coxeter diagram | Template:CDD or Template:CDD Template:CDD Template:CDD |
| Facets | {3,3,3,3,4}  h{4,3,3,3,3} 
|
| Vertex figure | t1{3,3,3,3,4} 
|
| Coxeter group | [4,3,3,3,31,1] [31,1,3,3,31,1] |
The 6-demicubic honeycomb or demihexeractic honeycube is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.
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 D6 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 E6 lattice and the 222 honeycomb.
The DTemplate:Sup sub lattice (also called DTemplate:Sup sub) can be constructed by the union of two D6 lattices. This packing is only a lattice for even dimensions. The kissing number is 25=32 (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 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 D6* lattice is 12 (2n for n≥5).[5] and its Voronoi tessellation is a trirectified 6-cubic honeycomb, Template:CDD, containing all birectified 6-orthoplex Voronoi cell, Template:CDD.[6]
Symmetry constructions
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of differened colors on the 64 6-demicube facets around each vertex.
| Coxeter group | Schläfli symbol | Coxeter-Dynkin diagram | Vertex figure Symmetry |
Facets/verf |
|---|---|---|---|---|
| = [31,1,3,3,3,4] = [1+,4,3,3,3,3,4] |
{31,1,3,3,3,4} = h{4,3,3,3,3,4} |
Template:CDD = Template:CDD | Template:CDD [3,3,3,3,4] |
64: 6-demicube 12: 6-orthoplex |
| = [31,1,3,31,1] = [1+,4,3,3,31,1] |
{31,1,3,3,31,1} | Template:CDD = Template:CDD | Template:CDD [33,1,1] |
32+32: 6-demicube 12: 6-orthoplex |
| = ([[4,3,3,3,4,2+]]) | ht0,5{4,3,3,3,4} | Template:CDD | 32+16+16: 6-demicube 12: 6-orthoplex |
Related honeycombs
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External links
- Template:GlossaryForHyperspace
- 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/D6.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/Ds6.html
- ↑ Conway (1998), p. 120
- ↑ Conway (1998), p. 466