Bennett's inequality: Difference between revisions

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!bgcolor=#e7dcc3 colspan=2|5-simplex honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||[[Uniform_6-polytope#Regular_and_uniform_honeycombs|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|Family||[[Simplectic honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{3<sup>[6]</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram]]s||{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
|-
|bgcolor=#e7dcc3|5-face types||[[hexateron|{3<sup>4</sub>}]] [[File:5-simplex t0.svg|30px]], [[rectified hexateron|t<sub>1</sub>{3<sup>4</sub>}]] [[File:5-simplex t1.svg|30px]]<br />[[birectified hexateron|t<sub>2</sub>{3<sup>4</sub>}]] [[File:5-simplex t2.svg|30px]]
|-
|bgcolor=#e7dcc3|4-face types||[[5-cell|{3<sup>3</sub>}]] [[File:4-simplex t0.svg|30px]], [[Rectified 5-cell|t<sub>1</sub>{3<sup>3</sub>}]] [[File:4-simplex t1.svg|30px]]
|-
|bgcolor=#e7dcc3|Cell types||[[tetrahedron|{3,3}]] [[File:3-simplex t0.svg|30px]], [[Octahedron|t<sub>1</sub>{3,3}]] [[File:3-simplex t1.svg|30px]]
|-
|bgcolor=#e7dcc3|Face types||[[triangle|{3}]] [[File:2-simplex t0.svg|30px]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[Stericated 5-simplex|t<sub>0,4</sub>{3<sup>4</sup>}]] [[File:5-simplex t04.svg|30px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||<math>{\tilde{A}}_5</math>×2, <[3<sup>[6]</sup>]>
|-
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
|}
In [[Five-dimensional space|five-dimensional]] [[Euclidean geometry]], the '''5-simplex honeycomb''' or '''hexateric honeycomb''' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]] or pentacomb). Each vertex is shared by 12 [[5-simplex]]es, 30 [[rectified 5-simplex]]es, and 20 [[birectified 5-simplex]]es. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.
 
== A5 lattice ==
This [[vertex arrangement]] is called the [[A5 lattice|A<sub>5</sub> lattice]] or '''5-simplex lattice'''. The 30 vertices of the [[stericated 5-simplex]] vertex figure represent the 30 roots of the <math>{\tilde{A}}_5</math> Coxeter group.<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A5.html</ref> It is the 5-dimensional case of a [[simplectic honeycomb]].
 
The A{{sup sub|2|5}} lattice can be constructed as the union of two A<sub>5</sub> lattices:
:
{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}} + {{CDD|node|split1|nodes|3ab|nodes|split2|node_1}}
 
The A{{sup sub|3|5}} is the union of three A<sub>5</sub> lattices:
:
{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}} + {{CDD|node|split1|nodes|3ab|nodes_10lru|split2|node}} + {{CDD|node|split1|nodes|3ab|nodes_01lr|split2|node}}.
 
The A{{sup sub|*|5}} lattice (also called A{{sup sub|6|5}}) is the union of six A<sub>5</sub> lattices, and is the dual [[vertex arrangement]] to the [[omnitruncated 5-simplex honeycomb]], and therefore the [[Voronoi cell]] of this lattice is an [[omnitruncated 5-simplex]].
:
{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}} +
{{CDD|node|split1|nodes_10lur|3ab|nodes|split2|node}} +
{{CDD|node|split1|nodes_01lr|3ab|nodes|split2|node}} +
{{CDD|node|split1|nodes|3ab|nodes_10lru|split2|node}} +
{{CDD|node|split1|nodes|3ab|nodes_01lr|split2|node}} +
{{CDD|node|split1|nodes|3ab|nodes|split2|node_1}} = dual of {{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}}
 
== Related polytopes and honeycombs ==
 
{{5-simplex honeycomb family}}
 
== Projection by folding ==
 
The ''5-simplex honeycomb'' can be projected into the 3-dimensional [[cubic honeycomb]] by a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] operation that maps two pairs of mirrors into each other, sharing the same [[vertex arrangement]]:
 
{|class=wikitable
|-
!<math>{\tilde{A}}_5</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
|-
!<math>{\tilde{C}}_3</math>
|{{CDD|node_1|4|node|3|node|4|node}}
|}
 
==See also==
Regular and uniform honeycombs in 5-space:
*[[5-cubic honeycomb]]
*[[5-demicube honeycomb]]
*[[Truncated 5-simplex honeycomb]]
*[[Omnitruncated 5-simplex honeycomb]]
 
==Notes==
{{reflist}}
 
== References ==
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
* '''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 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
{{Honeycombs}}
 
[[Category:Honeycombs (geometry)]]
[[Category:6-polytopes]]

Revision as of 22:59, 6 February 2014

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