6-demicubic honeycomb
6-demicubic honeycomb | |
---|---|
(No image) | |
Type | Uniform honeycomb |
Family | Alternated hypercube honeycomb |
Schläfli symbol | h{4,3,3,3,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Facets | {3,3,3,3,4} ![]() h{4,3,3,3,3} ![]() |
Vertex figure | t1{3,3,3,3,4} 25px |
Coxeter group | ![]() ![]() |
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.
Contents
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 D+
6 lattice (also called D2
6) 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 D*
6 lattice (also called D4
6 and C2
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 D6* lattice is 12 (2n for n≥5).[5] and its Voronoi tessellation is a trirectified 6-cubic honeycomb, , containing all birectified 6-orthoplex Voronoi cell,
.[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 |
---|---|---|---|---|
![]() = [1+,4,3,3,3,3,4] |
= h{4,3,3,3,3,4} | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3,3,4] |
64: 6-demicube 12: 6-orthoplex |
![]() = [1+,4,3,3,31,1] |
= h{4,3,3,3,31,1} | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [33,1,1] |
32+32: 6-demicube 12: 6-orthoplex |
![]() |
ht0,6{4,3,3,3,3,4} | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32+16+16: 6-demicube 12: 6-orthoplex |
Related honeycombs
This honeycomb is one of 41 uniform honycombs 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 |
[31,1,3,3,31,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[[31,1,3,3,31,1]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
<[31,1,3,3,31,1]> ↔ [31,1,3,3,3,4] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ↔ ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
|
<2[31,1,3,3,31,1]> ↔ [4,3,3,3,3,4] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ↔ ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
|
[<2[31,1,3,3,31,1]>] ↔ [[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 [1]
- ↑ 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
- Olshevsky, George, Half measure polytope at Glossary for Hyperspace.
- 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]
- Lua error in package.lua at line 80: module 'strict' not found.
Fundamental convex regular and uniform honeycombs in dimensions 2–10 | |||||
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Family | ![]() |
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Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
Uniform 5-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
Uniform 6-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
Uniform 7-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
Uniform 8-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
Uniform 9-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
Uniform n-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |