6-demicubic honeycomb

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6-demicubic honeycomb
(No image)
Type Uniform honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol h{4,3,3,3,3,4}
Coxeter diagram CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png or CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
Facets {3,3,3,3,4} 6-cube t5.svg
h{4,3,3,3,3} 6-demicube t0 D6.svg
Vertex figure t1{3,3,3,3,4} Rectified hexacross.svg
Coxeter group {\tilde{B}}_6 [4,3,3,3,31,1]
{\tilde{D}}_6 [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[edit]

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]

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png

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.

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes 01rd.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png.

The kissing number of the D6* lattice is 12 (2n for n≥5).[5] and its Voronoi tessellation is a trirectified 6-cubic honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png, containing all birectified 6-orthoplex Voronoi cell, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.[6]

Symmetry constructions[edit]

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
{\tilde{B}}_6 = [31,1,3,3,3,4]
= [1+,4,3,3,3,3,4]
= h{4,3,3,3,3,4} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[3,3,3,4]
64: 6-demicube
12: 6-orthoplex
{\tilde{D}}_6 = [31,1,3,31,1]
= [1+,4,3,3,31,1]
= h{4,3,3,3,31,1} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[33,1,1]
32+32: 6-demicube
12: 6-orthoplex
{\tilde{C}}_6 = [[(4,3,3,3,4,2+)]] ht0,6{4,3,3,3,3,4} CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png 32+16+16: 6-demicube
12: 6-orthoplex

Related honeycombs[edit]

This honeycomb is one of 41 uniform honycombs constructed by the {\tilde{D}}_6 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 {\tilde{B}}_6 and {\tilde{C}}_6 constructions:

Extended
symmetry
Extended
diagram
Order Honeycombs
[31,1,3,3,31,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png ×1 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
[[31,1,3,3,31,1]] CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png ×2 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
<[31,1,3,3,31,1]>
= [31,1,3,3,3,4]
CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.pngCDel split1.pngCDel nodeab c6.png
= CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.pngCDel 3.pngCDel node c6.pngCDel 4.pngCDel node.png
×2 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png,

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png

<<[31,1,3,3,31,1]>>
= [4,3,3,3,3,4]
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel split1.pngCDel nodeab c5.png
= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.pngCDel 4.pngCDel node.png
×4 CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png,CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png,

CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png,CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png,

CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png

[<<[31,1,3,3,31,1]>>]
= [[4,3,3,3,3,4]]
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png
= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.png
×8 CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png,

CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png

See also[edit]

Notes[edit]

  1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D6.html
  2. ^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
  3. ^ Conway (1998), p. 119
  4. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds6.html
  5. ^ Conway (1998), p. 120
  6. ^ Conway (1998), p. 466

External links[edit]

  • 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]
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.