7-demicubic honeycomb

The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb.

7-demicubic honeycomb
(No image)
TypeUniform 7-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbolh{4,3,3,3,3,3,4}
h{4,3,3,3,3,31,1}
ht0,7{4,3,3,3,3,3,4}
Coxeter-Dynkin diagram =
=
Facets{3,3,3,3,3,4}
h{4,3,3,3,3,3}
Vertex figureRectified 7-orthoplex
Coxeter group [4,3,3,3,3,31,1]
, [31,1,3,3,3,31,1]

It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h{4,3,3,3,3,3} and the alternated vertices create 7-orthoplex {3,3,3,3,3,4} facets.

D7 lattice

The vertex arrangement of the 7-demicubic honeycomb is the D7 lattice.[1] The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice.[2] The best known is 126, from the E7 lattice and the 331 honeycomb.

The D+
7 packing (also called D2
7) can be constructed by the union of two D7 lattices. The D+
n packings form lattices only in even dimensions. The kissing number is 26=64 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]

∪

The D*
7 lattice (also called D4
7 and C2
7) can be constructed by the union of all four 7-demicubic lattices:[4] 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*
7 lattice is 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb, , containing all with tritruncated 7-orthoplex, Voronoi cells.[5]

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 128 7-demicube facets around each vertex.

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry		 Facets/verf
= [31,1,3,3,3,3,4]
= [1+,4,3,3,3,3,3,4]		h{4,3,3,3,3,3,4} =
[3,3,3,3,3,4]		 128: 7-demicube
14: 7-orthoplex
= [31,1,3,3,31,1]
= [1+,4,3,3,3,31,1]		h{4,3,3,3,3,31,1} =
[35,1,1]		 64+64: 7-demicube
14: 7-orthoplex
2×½ = [[(4,3,3,3,3,4,2<sup>+</sup>)]]ht0,7{4,3,3,3,3,3,4} 64+32+32: 7-demicube
14: 7-orthoplex

See also

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

Notes
  1. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D7.html
  2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
  3. Conway (1998), p. 119
  4. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds7.html
  5. Conway (1998), p. 466
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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