8-demicubic honeycomb

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8-demicubic honeycomb
(No image)
TypeUniform 8-space honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbolh{4,3,3,3,3,3,3,4}
Coxeter-Dynkin diagram or

Facets{3,3,3,3,3,3,4}
h{4,3,3,3,3,3,3}
Vertex figureRectified octacross
Coxeter group{{\tilde {B}}}_{8} [4,3,3,3,3,3,31,1]
{{\tilde {D}}}_{8} [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.

{{\tilde {E}}}_{8} contains {{\tilde {D}}}_{8} as a subgroup of index 270.[3] Both {{\tilde {E}}}_{8} and {{\tilde {D}}}_{8} can be seen as affine extensions of D_{8} from different nodes:

The D+
8 lattice (also called D2
8) can be constructed by the union of two D8 lattices. 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).[4] 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:[5] 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).[6] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, , containing all trirectified 8-orthoplex Voronoi cell, .[7]

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 (2011) p.177
  4. ↑ Conway (1998), p. 119
  5. ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds8.html
  6. ↑ Conway (1998), p. 120
  7. ↑ 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, Manuscript, (2011)
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9. 

External links

Fundamental convex regular and uniform honeycombs in dimensions 2–11
Family {{\tilde {A}}}_{{n-1}} {{\tilde {C}}}_{{n-1}} {{\tilde {B}}}_{{n-1}} {{\tilde {D}}}_{{n-1}} {{\tilde {G}}}_{2} / {{\tilde {F}}}_{4} / {{\tilde {E}}}_{{n-1}}
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
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