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+)]]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. "The Lattice D7".
  2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
  3. Conway (1998), p. 119
  4. "The Lattice D7".
  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
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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