Bennett's inequality: Difference between revisions

Revision as of 15:10, 27 November 2013

5-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3^[6]}
Coxeter–Dynkin diagrams	Template:CDD
5-face types	{3^4} , t₁{3^4} t₂{3^4}
4-face types	{3^3} , t₁{3^3}
Cell types	{3,3} , t₁{3,3}
Face types	{3}
Vertex figure	t_0,4{3⁴}
Coxeter groups	${\tilde {A}}_{5}$ ×2, <[3^[6]]>
Properties	vertex-transitive

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. 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 or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the ${\tilde {A}}_{5}$ Coxeter group.^[1] It is the 5-dimensional case of a simplectic honeycomb.

The ATemplate:Sup sub lattice can be constructed as the union of two A₅ lattices:

Template:CDD + Template:CDD

The ATemplate:Sup sub is the union of three A₅ lattices:

Template:CDD + Template:CDD + Template:CDD.

The ATemplate:Sup sub lattice (also called ATemplate:Sup sub) is the union of six A₅ 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.

Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD = dual of Template:CDD

Related polytopes and honeycombs

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Projection by folding

The 5-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

${\tilde {A}}_{5}$	Template:CDD
${\tilde {C}}_{3}$	Template:CDD

Notes

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References

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 [1]
- (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]

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↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A5.html

[1] ttp://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A5.html

[1]

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+{| class="wikitable" align="right" style="margin-left:10px" width="320"
+!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]]

Bennett's inequality: Difference between revisions

Revision as of 15:10, 27 November 2013

Contents

A5 lattice

Related polytopes and honeycombs

Projection by folding

See also

Notes

References

Navigation menu

Bennett's inequality: Difference between revisions

Revision as of 15:10, 27 November 2013

A5 lattice

Related polytopes and honeycombs

Projection by folding

See also

Notes

References

Navigation menu

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