8-demicubic honeycomb

8-demicubic honeycomb
(No image)
TypeUniform 8-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbolh{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 figureRectified 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

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

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<sup>+</sup>)]]ht0,8{4,3,3,3,3,3,3,4} 128+64+64: 8-demicube
16: 8-orthoplex

See also

Notes

  1. ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D8.html
  2. ↑ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
  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. ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds8.html
  7. ↑ Conway (1998), p. 120
  8. ↑ Conway (1998), p. 466

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
    • pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}, ...
  • 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
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • N.W. Johnson: Geometries and Transformations, (2018)
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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