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5-demicubic honeycomb

Type of uniform space-filling tessellation

Type of uniform space-filling tessellation

Demipenteractic honeycomb
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Type
Family
Schläfli symbols
Coxeter diagrams
Facets
Vertex figure
Coxeter group

The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.

It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h{4,3,3,3} and the alternated vertices create 5-orthoplex {3,3,3,4} facets.

D5 lattice

The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 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 D packing (also called D) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n−1 for n8). : ∪

The D lattice (also called D and C) can be constructed by the union of all four 5-demicubic lattices: It is also the 5-dimensional body centered cubic, the union of two 5-cube honeycombs in dual positions. : ∪ ∪ ∪ = ∪ .

The kissing number of the D lattice is 10 (2n for n≥5) and its Voronoi tessellation is a tritruncated 5-cubic honeycomb, , containing all bitruncated 5-orthoplex, Voronoi cells.

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 32 5-demicube facets around each vertex.

Coxeter groupSchläfli symbolCoxeter-Dynkin diagramVertex figureSymmetryFacets/verf
{\tilde{B}}_5 = [31,1,3,3,4]= [1+,4,3,3,4]h{4,3,3,3,4}=[3,3,3,4]32: 5-demicube10: 5-orthoplex
{\tilde{D}}_5 = [31,1,3,31,1]= [1+,4,3,31,1]h{4,3,3,31,1}=[32,1,1]16+16: 5-demicube10: 5-orthoplex
2×½{\tilde{C}}_5 = (4,3,3,3,4,2+)ht0,5{4,3,3,3,4}16+8+8: 5-demicube10: 5-orthoplex

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
    • 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

References

  1. "The Lattice D5".
  2. Conway (1998), p. 119
  3. "The Lattice D5".
  4. Conway (1998), p. 120
  5. Conway (1998), p. 466
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honeycombs-(geometry) 6-polytopes
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