8-demicubic honeycomb

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8-demicubic honeycomb
(No image)
Type Uniform 8-honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol h{4,3,3,3,3,3,3,4}
Coxeter diagrams =
=
Facets {3,3,3,3,3,3,4}
h{4,3,3,3,3,3,3}
Vertex figure Rectified 8-orthoplex
Coxeter group [4,3,3,3,3,3,31,1]
[31,1,3,3,3,3,31,1]

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb.

It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h{4,3,3,3,3,3,3} and the alternated vertices create 8-orthoplex {3,3,3,3,3,3,4} facets .

D8 lattice

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The vertex arrangement of the 8-demicubic honeycomb is the D8 lattice.[1] The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice.[2] The best known is 240, from the E8 lattice and the 521 honeycomb.

  contains   as a subgroup of index 270.[3] Both   and   can be seen as affine extensions of   from different nodes:  

The D+
8
lattice (also called D2
8
) can be constructed by the union of two D8 lattices.[4] This packing is only a lattice for even dimensions. The kissing number is 240. (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[5] It is identical to the E8 lattice. At 8-dimensions, the 240 contacts contain both the 27=128 from lower dimension contact progression (2n-1), and 16*7=112 from higher dimensions (2n(n-1)).

                           =                .

The D*
8
lattice (also called D4
8
and C2
8
) can be constructed by the union of all four D8 lattices:[6] It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

                                                     =                   .

The kissing number of the D*
8
lattice is 16 (2n for n≥5).[7] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb,          , containing all trirectified 8-orthoplex Voronoi cell,                .[8]

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 256 8-demicube facets around each vertex.

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
  = [31,1,3,3,3,3,3,4]
= [1+,4,3,3,3,3,3,3,4]
h{4,3,3,3,3,3,3,4}                 =                                  
[3,3,3,3,3,3,4]
256: 8-demicube
16: 8-orthoplex
  = [31,1,3,3,3,31,1]
= [1+,4,3,3,3,3,31,1]
h{4,3,3,3,3,3,31,1}               =                              
[36,1,1]
128+128: 8-demicube
16: 8-orthoplex
2×½  = [[(4,3,3,3,3,3,4,2+)]] ht0,8{4,3,3,3,3,3,3,4}            128+64+64: 8-demicube
16: 8-orthoplex

See also

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Notes

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  1. ^ "The Lattice D8".
  2. ^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
  3. ^ Johnson (2015) p.177
  4. ^ Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
  5. ^ Conway (1998), p. 119
  6. ^ "The Lattice D8".
  7. ^ Conway (1998), p. 120
  8. ^ Conway (1998), p. 466

References

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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