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6-demicubic honeycomb

6-demicubic honeycomb
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Type
Family
Schläfli symbol
Coxeter diagram
Facets
Vertex figure
Coxeter group

The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.

It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h{4,3,3,3,3} and the alternated vertices create 6-orthoplex {3,3,3,3,4} facets.

D6 lattice

The vertex arrangement of the 6-demicubic honeycomb is the D6 lattice. The 60 vertices of the rectified 6-orthoplex vertex figure of the 6-demicubic honeycomb reflect the kissing number 60 of this lattice.Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai https://books.google.com/books?id=upYwZ6cQumoC&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19 The best known is 72, from the E6 lattice and the 222 honeycomb.

The D lattice (also called D) can be constructed by the union of two D6 lattices. This packing is only a lattice for even dimensions. The kissing number is 25=32 (2n-1 for n8). : ∪

The D lattice (also called D and C) can be constructed by the union of all four 6-demicubic lattices: It is also the 6-dimensional body centered cubic, the union of two 6-cube honeycombs in dual positions. : ∪ ∪ ∪ = ∪ .

The kissing number of the D6* lattice is 12 (2n for n≥5). and its Voronoi tessellation is a trirectified 6-cubic honeycomb, , containing all birectified 6-orthoplex Voronoi cell, .

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 64 6-demicube facets around each vertex.

Coxeter groupSchläfli symbolCoxeter-Dynkin diagramVertex figureSymmetryFacets/verf
{\tilde{B}}_6 = [31,1,3,3,3,4]= [1+,4,3,3,3,3,4]h{4,3,3,3,3,4}=[3,3,3,4]64: 6-demicube12: 6-orthoplex
{\tilde{D}}_6 = [31,1,3,31,1]= [1+,4,3,3,31,1]h{4,3,3,3,31,1}=[33,1,1]32+32: 6-demicube12: 6-orthoplex
½{\tilde{C}}_6 = (4,3,3,3,4,2+)ht0,6{4,3,3,3,3,4}32+16+16: 6-demicube12: 6-orthoplex

Notes

References

  1. "The Lattice D6".
  2. Conway (1998), p. 119
  3. "The Lattice D6".
  4. Conway (1998), p. 120
  5. Conway (1998), p. 466
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honeycombs-(geometry) 7-polytopes
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