6-demicubic honeycomb

Summary

6-demicubic honeycomb
(No image)
Type Uniform 6-honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol h{4,3,3,3,3,4}
h{4,3,3,3,31,1}
ht0,6{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 [4,3,3,3,31,1]
[31,1,3,3,31,1]

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.

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

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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

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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
  = [31,1,3,3,3,4]
= [1+,4,3,3,3,3,4]
h{4,3,3,3,3,4}             =                        
[3,3,3,4]
64: 6-demicube
12: 6-orthoplex
  = [31,1,3,31,1]
= [1+,4,3,3,31,1]
h{4,3,3,3,31,1}           =                      
[33,1,1]
32+32: 6-demicube
12: 6-orthoplex
½  = [[(4,3,3,3,4,2+)]] ht0,6{4,3,3,3,3,4}          32+16+16: 6-demicube
12: 6-orthoplex
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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
[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

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Notes

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  1. ^ "The Lattice D6".
  2. ^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
  3. ^ Conway (1998), p. 119
  4. ^ "The Lattice D6".
  5. ^ Conway (1998), p. 120
  6. ^ Conway (1998), p. 466
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  • 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.
Space Family           /   /  
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21