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7-demicubic honeycomb#D7 lattice

{{Short description|Uniform 7-Honeycomb}}

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!bgcolor=#e7dcc3 colspan=2|7-demicubic honeycomb

bgcolor=#ffffff align=center colspan=2|(No image)
bgcolor=#e7dcc3|TypeUniform 7-honeycomb
bgcolor=#e7dcc3|FamilyAlternated hypercube honeycomb
bgcolor=#e7dcc3|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}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|4|node}} = {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|4|node}}
{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|split1|nodes}}
{{CDD|label2|branch_hh|4a4b|nodes|3ab|nodes|3ab|branch}}
bgcolor=#e7dcc3|Facets{3,3,3,3,3,4}
h{4,3,3,3,3,3}
bgcolor=#e7dcc3|Vertex figureRectified 7-orthoplex
bgcolor=#e7dcc3|Coxeter group{\tilde{B}}_7 [4,3,3,3,3,31,1]
{\tilde{D}}_7, [31,1,3,3,3,31,1]

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.

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.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D7.html|title = The Lattice D7}} The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 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 126, from the E7 lattice and the 331 honeycomb.

The D{{sup sub|+|7}} packing (also called D{{sup sub|2|7}}) can be constructed by the union of two D7 lattices. The D{{sup sub|+|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).Conway (1998), p. 119

:{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|split1|nodes}} ∪ {{CDD|nodes|split2|node|3|node|3|node|3|node|split1|nodes_10lu}}

The D{{sup sub|*|7}} lattice (also called D{{sup sub|4|7}} and C{{sup sub|2|7}}) can be constructed by the union of all four 7-demicubic lattices:{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds7.html|title=The Lattice D7}} It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

:{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|split1|nodes}} ∪ {{CDD|nodes_01rd|split2|node|3|node|3|node|3|node|split1|nodes}} ∪ {{CDD|nodes|split2|node|3|node|3|node|3|node|split1|nodes_10lu}} ∪ {{CDD|nodes|split2|node|3|node|3|node|3|node|split1|nodes_01ld}} = {{CDD|nodes_10r|4a4b|nodes|3ab|nodes|3ab|branch}} ∪ {{CDD|nodes_01r|4a4b|nodes|3ab|nodes|3ab|branch}}.

The kissing number of the D{{sup sub|*|7}} lattice is 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb, {{CDD|branch_11|3ab|nodes|3ab|nodes|4a4b|nodes}}, containing all with tritruncated 7-orthoplex, {{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node}} Voronoi cells.Conway (1998), p. 466

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.

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!Coxeter group

!Schläfli symbol

!Coxeter-Dynkin diagram

!Vertex figure
Symmetry

!Facets/verf

{\tilde{B}}_7 = [31,1,3,3,3,3,4]
= [1+,4,3,3,3,3,3,4]		h{4,3,3,3,3,3,4}{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|4|node}}{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|4|node}}
[3,3,3,3,3,4]		128: 7-demicube
14: 7-orthoplex
{\tilde{D}}_7 = [31,1,3,3,31,1]
= [1+,4,3,3,3,31,1]		h{4,3,3,3,3,31,1}{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|split1|nodes}}{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|split1|nodes}}
[35,1,1]		64+64: 7-demicube
14: 7-orthoplex
2×½{\tilde{C}}_7 = (4,3,3,3,3,4,2+)ht0,7{4,3,3,3,3,3,4}{{CDD|label2|branch_hh|3ab|nodes|4a4b|nodes|3ab|branch}}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}} [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]
  • {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |publisher=Springer |edition=3rd |isbn=0-387-98585-9 |url-access=registration |url=https://archive.org/details/spherepackingsla0000conw_b8u0 }}

Notes

{{reflist}}

External links

{{Honeycombs}}

{{DEFAULTSORT:Demihepteractic Honeycomb}}

Category:Honeycombs (geometry)

Category:8-polytopes