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5-demicubic honeycomb#D5 lattice

{{Short description|Type of uniform space-filling tessellation}}

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!bgcolor=#e7dcc3 colspan=2|Demipenteractic honeycomb

bgcolor=#ffffff align=center colspan=2|(No image)
bgcolor=#e7dcc3|TypeUniform 5-honeycomb
bgcolor=#e7dcc3|FamilyAlternated hypercubic honeycomb
bgcolor=#e7dcc3|Schläfli symbolsh{4,3,3,3,4}
h{4,3,3,31,1}
ht0,5{4,3,3,3,4}
h{4,3,3,4}h{∞}
h{4,3,31,1}h{∞}
ht0,4{4,3,3,4}h{∞}
h{4,3,4}h{∞}h{∞}
h{4,31,1}h{∞}h{∞}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node}}


{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|split1|nodes}}


{{CDD|label2|branch_hh|4a4b|nodes|3ab|branch}}


{{CDD|node_h|4|node|3|node|3|node|4|node|2|node_h|infin|node}}


{{CDD|node_h|4|node|3|node|split1|nodes|2|node_h|infin|node}}


{{CDD|node_h|4|node|3|node|3|node|4|node_h|2|node_h|infin|node}}


{{CDD|node_h|4|node|3|node|4|node|2|node_h|infin|node|2|node_h|infin|node}}


{{CDD|node_h|4|node|split1|nodes|2|node_h|infin|node|2|node_h|infin|node}}


{{CDD|node_h|4|node|3|node|4|node_h|2|node_h|infin|node|2|node_h|infin|node}}


{{CDD|node_h|4|node|4|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}


{{CDD|node_h|4|node|4|node_h|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}


{{CDD|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}

bgcolor=#e7dcc3|Facets{3,3,3,4} 25px
h{4,3,3,3} 25px
bgcolor=#e7dcc3|Vertex figuret1{3,3,3,4} 25px
bgcolor=#e7dcc3|Coxeter group{\tilde{B}}_5 [4,3,3,31,1]
{\tilde{D}}_5 [31,1,3,31,1]

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.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.html|title = The Lattice D5}} 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{{sup sub|+|5}} packing (also called D{{sup sub|2|5}}) 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 n<8, 240 for n=8, and 2n(n−1) for n>8).Conway (1998), p. 119

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

The D{{sup sub|*|5}}{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html|title=The Lattice D5}} lattice (also called D{{sup sub|4|5}} and C{{sup sub|2|5}}) can be constructed by the union of all four 5-demicubic lattices:Conway (1998), p. 120 It is also the 5-dimensional body centered cubic, the union of two 5-cube honeycombs in dual positions.

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

The kissing number of the D{{sup sub|*|5}} lattice is 10 (2n for n≥5) and its Voronoi tessellation is a tritruncated 5-cubic honeycomb, {{CDD|branch_11|3ab|nodes|4a4b|nodes}}, containing all bitruncated 5-orthoplex, {{CDD|node|4|node|3|node_1|3|node_1|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 32 5-demicube facets around each vertex.

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!Coxeter group

!Schläfli symbol

!Coxeter-Dynkin diagram

!Vertex figure
Symmetry

!Facets/verf

{\tilde{B}}_5 = [31,1,3,3,4]
= [1+,4,3,3,4]		h{4,3,3,3,4}{{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node}}{{CDD|node|3|node_1|3|node|3|node|4|node}}
[3,3,3,4]		32: 5-demicube
10: 5-orthoplex
{\tilde{D}}_5 = [31,1,3,31,1]
= [1+,4,3,31,1]		h{4,3,3,31,1}{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|split1|nodes}}{{CDD|node|3|node_1|3|node|split1|nodes}}
[32,1,1]		16+16: 5-demicube
10: 5-orthoplex
2×½{\tilde{C}}_5 = (4,3,3,3,4,2+)ht0,5{4,3,3,3,4}{{CDD|label2|branch_hh|4a4b|nodes|3ab|branch}}16+8+8: 5-demicube
10: 5-orthoplex

Related honeycombs

{{D5 honeycombs}}

See also

Regular and uniform honeycombs in 5-space:

References

{{reflist}}

  • 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 }}

External links

{{Honeycombs}}

{{DEFAULTSORT:Demipenteractic Honeycomb}}

Category:Honeycombs (geometry)

Category:6-polytopes