en>Qwertyytrewqqwerty: Disambiguating links to Sum of squares using DisamAssist.

2013-12-31T14:32:24Z

Disambiguating links to Sum of squares using DisamAssist.

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{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|6-demicubic honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||[[Uniform_polyexon#Regular_and_uniform_honeycombs|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|Family||[[Alternated hypercube honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,3,3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|4|node}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|4|node}} {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|4|node_h}} {{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|[[Facet (geometry)|Facets]]||[[hexacross|{3,3,3,3,4}]] [[File:6-cube t5.svg|25px]] [[demihexeract|h{4,3,3,3,3}]] [[File:6-demicube t0 D6.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Rectified hexacross|t1{3,3,3,3,4}]] [[File:Rectified hexacross.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{B}}_6</math> [4,3,3,3,31,1] <math>{\tilde{D}}_6</math> [31,1,3,3,31,1]
|}
The '''6-demicubic honeycomb''' or '''demihexeractic honeycube''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 6-space. It is constructed as an [[Alternation (geometry)|alternation]] of the regular [[6-cube honeycomb]].

It is composed of two different types of [[Facet (mathematics)|facet]]s. The [[6-cube]]s become alternated into [[6-demicube]]s 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'''.<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D6.html</ref> The 60 vertices of the [[rectified 6-orthoplex]] [[vertex figure]] of the ''6-demicubic honeycomb'' reflect the [[kissing number]] 60 of this lattice.<ref>''Sphere packings, lattices, and groups'', by [[John Horton Conway]], Neil James Alexander Sloane, Eiichi Bannai
[http://books.google.com/books?id=upYwZ6cQumoC&lpg=PP1&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19#v=onepage&q=&f=false]</ref> The best known is 72, from the [[E6 lattice|E6 lattice]] and the [[2 22 honeycomb|222 honeycomb]].

The D{{sup sub|+|6}} lattice (also called D{{sup sub|2|6}}) 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 n<8, 240 for n=8, and 2n(n-1) for n>8).<ref>Conway (1998), p. 119</ref>
:{{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}} + {{CDD|nodes|split2|node|3|node|3|node|split1|nodes_10lu}}

The D{{sup sub|*|6}} lattice (also called D{{sup sub|4|6}} and C{{sup sub|2|6}}) can be constructed by the union of all four 6-demicubic lattices:<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds6.html</ref> It is also the 6-dimensional [[body centered cubic]], the union of two [[6-cube honeycomb]]s in dual positions.
:{{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}} + {{CDD|nodes_01rd|split2|node|3|node|3|node|split1|nodes}} + {{CDD|nodes|split2|node|3|node|3|node|split1|nodes_10lu}} + {{CDD|nodes|split2|node|3|node|3|node|split1|nodes_01ld}} = {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|4|node}} + {{CDD|node|4|node|3|node|3|node|3|node|3|node|4|node_1}}.

The [[kissing number]] of the D6* lattice is 12 (''2n'' for n≥5).<ref>Conway (1998), p. 120</ref> and its [[Voronoi tessellation]] is a [[trirectified 6-cubic honeycomb]], {{CDD|node_1|split1|nodes|3ab|nodes|4a4b|nodes}}, containing all [[birectified 6-orthoplex]] [[Voronoi cell]], {{CDD|node|4|node|3|node|3|node_1|3|node|3|node}}.<ref>Conway (1998), p. 466</ref>

== Symmetry constructions ==

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

{|class='wikitable'
![[Coxeter group]]
![[Schläfli symbol]]
![[Coxeter-Dynkin diagram]]
![[Vertex figure]] Symmetry
![[Facet (geometry)|Facets]]/verf
|-
|<math>{\tilde{B}}_6</math> = [31,1,3,3,3,4] = [1+,4,3,3,3,3,4]||{31,1,3,3,3,4} = h{4,3,3,3,3,4}||{{CDD|nodes_10ru|split2|node|3|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}} [3,3,3,3,4]
||64: [[6-demicube]] 12: [[6-orthoplex]]
|-
|<math>{\tilde{D}}_6</math> = [31,1,3,31,1] = [1+,4,3,3,31,1]||{31,1,3,3,31,1}||{{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|split1|nodes}}||{{CDD|node|3|node_1|3|node|3|node|split1|nodes}} [33,1,1]
||32+32: [[6-demicube]] 12: [[6-orthoplex]]
|-
|<math>{\tilde{C}}_6</math> = ([[4,3,3,3,4,2+]])||ht0,5{4,3,3,3,4}||{{CDD|node_h|4|node|3|node|3|node|3|node|3|node|4|node_h}}||
||32+16+16: [[6-demicube]] 12: [[6-orthoplex]]
|}

== Related honeycombs==
{{D6_honeycombs}}

==See also ==
*[[6-cubic honeycomb]]

==Notes==
{{reflist}}

== External links ==
*{{GlossaryForHyperspace | anchor=half | title=Half measure polytope }}
* '''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 |edition=3rd |isbn=0-387-98585-9}}
{{Honeycombs}}

{{DEFAULTSORT:Demihexeractic Honeycomb}}
[[Category:Honeycombs (geometry)]]
[[Category:7-polytopes]]

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en>Qwertyytrewqqwerty: Disambiguating links to Sum of squares using DisamAssist.