Askey–Wilson polynomials: Difference between revisions

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{{DISPLAYTITLE:3<sub> 31</sub> honeycomb}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|'''3<sub>31</sub>''' honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(no image)
|-
|bgcolor=#e7dcc3|Type||[[Uniform_polyzetton#Regular_and_uniform_honeycombs|Uniform tessellation]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3,3<sup>3,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| '''3<sub>31</sub>'''
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]|| {{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|7-face types||'''[[3 21 polytope|3<sub>21</sub>]]''' [[File:E7 graph.svg|25px]]<BR>[[7-simplex|{3<sup>6</sup>}]] [[Image:7-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|6-face types||'''[[2 21 polytope|2<sub>21</sub>]]'''[[Image:E6 graph.svg|25px]]<BR>[[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-face types||'''[[5-orthoplex|2<sub>11</sub>]]'''[[Image:Cross graph 5.svg|25px]]<BR>[[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-face type||[[5-cell|{3<sup>3</sup>}]][[Image:4-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cell type||[[tetrahedron|{3<sup>2</sup>}]][[Image:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Face type||[[triangle|{3}]][[Image:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Face figure||'''[[rectified 5-simplex|0<sub>31</sub>]]''' [[File:5-simplex t1.svg|25px]]
|-
|bgcolor=#e7dcc3|Edge figure||'''[[6-demicube|1<sub>31</sub>]]''' [[File:6-demicube.svg|25px]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[2 31 polytope|2<sub>31</sub>]] [[File:Gosset 2 31 polytope.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{E}}_7</math>, [3<sup>3,3,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
|}
In 7-dimensional [[geometry]], the '''3<sub>31</sub> honeycomb''' is a uniform honeycomb, also given by [[Schlafli symbol]] {3,3,3,3<sup>3,1</sup>} and is composed of '''[[3 21 polytope|3<sub>21</sub>]]''' and [[7-simplex]] [[Facet (geometry)|facets]], with 56 and 576 of them respectively around each vertex.


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==Construction==
 
It is created by a [[Wythoff construction]] upon a set of 8 [[hyperplane]] mirrors in 7-dimensional space.
 
The facet information can be extracted from its [[Coxeter-Dynkin diagram]].
: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
 
Removing the node on the short branch leaves the [[6-simplex]] facet:
: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
 
Removing the node on the end of the 3-length branch leaves the '''[[3 21 polytope|3<sub>21</sub>]]''' facet:
: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
 
The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes '''[[2 31 polytope|2<sub>31</sub>]]''' polytope.
: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
 
The [[edge figure]] is determined by removing the ringed node and ringing the neighboring node. This makes [[6-demicube]] ('''1<sub>31</sub>''').
: {{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
 
The [[face figure]] is determined by removing the ringed node and ringing the neighboring node. This makes [[rectified 5-simplex]] ('''0<sub>31</sub>''').
: {{CDD|branch_10|3a|nodea|3a|nodea|3a|nodea}}
 
The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes [[tetrahedral prism]]&nbsp;{}&times;{3,3}.
: {{CDD|node_1|2|node_1|3|node|3|node}}
 
== Kissing number ==
 
Each vertex of this tessellation is the center of a 6-sphere in the densest known [[sphere packing|packing]] in 7 dimensions; its [[kissing number]] is 126, represented by the vertices of its [[vertex figure]] [[2 31 polytope|2<sub>31</sub>]].
 
== E7 lattice ==
<math>{\tilde{E}}_7</math> contains <math>{\tilde{A}}_7</math> as a subgroup of index 144.<ref>N.W. Johnson: ''Geometries and Transformations'', Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p 177</ref> Both <math>{\tilde{E}}_7</math> and <math>{\tilde{A}}_7</math> can be seen as affine extension from <math>A_7</math> from different nodes: [[File:Affine_A7_E7_relations.png]]
 
The [[vertex arrangement]] of 3<sub>31</sub> is called the '''E<sub>7</sub> lattice'''.<ref>http://www2.research.att.com/~njas/lattices/E7.html</ref> The E<sub>7</sub> lattice can also be expressed as a union of the vertices of two A<sub>7</sub> lattices, also called A<sub>7</sub><sup>2</sup>:
:{{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_10l}} = {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}} + {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_1}}
 
The '''E<sub>7</sub><sup>*</sup> lattice''' (also called E<sub>7</sub><sup>2</sup>)<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html</ref> has double the symmetry, represented by [[3,3<sup>3,3</sup>]]. The [[Voronoi cell]] of the E<sub>7</sub><sup>*</sup> lattice is the [[1 32 polytope|1<sub>32</sub>]] polytope, and [[voronoi tessellation]] the [[1 33 honeycomb|1<sub>33</sub> honeycomb]].<ref>[http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices], Edward Pervin</ref> The '''E<sub>7</sub><sup>*</sup> lattice''' is constructed by 2 copies of the E<sub>7</sub> lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A<sub>7</sub><sup>*</sup> lattices, also called A<sub>7</sub><sup>4</sup>:
: {{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_10l}} + {{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_01l}} = {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}} + {{CDD|node|split1|nodes|3ab|nodes_10lr|3ab|nodes|split2|node}} + {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_1}}  + {{CDD|node|split1|nodes|3ab|nodes_01lr|3ab|nodes|split2|node}} = dual of {{CDD|node_1|3|node|split1|nodes|3ab|nodes|3ab|nodes}}.
 
== Related honeycombs ==
 
It is in a dimensional series of uniform polytopes and honeycombs, expressed by [[Coxeter]] as 3<sub>k1</sub> series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral [[hosohedron]].
{{3_k1_polytopes}}
 
== See also ==
* [[8-polytope]]
* [[1 33 honeycomb|1<sub>33</sub> honeycomb]]
 
== References ==
{{reflist}}
* [[H. S. M. Coxeter]], ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
* [[Harold Scott MacDonald Coxeter|Coxeter]] ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
* '''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] [http://books.google.com/books?id=fUm5Mwfx8rAC&lpg=PP1&dq=Coxeter&pg=PP1#v=onepage&q&f=false GoogleBook]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3&ndash;45]
* [[R. T. Worley]], ''The Voronoi Region of E7*''.  SIAM J. Disc. Math., 1.1 (1988), 134-141.
*{{Cite book| first = John H. | last = Conway | authorlink = John Horton Conway | coauthors  = [[Neil Sloane|Sloane, Neil J. A.]] | year = 1998 | title = Sphere Packings, Lattices and Groups | edition = (3rd ed.) | publisher = Springer-Verlag | location = New York | isbn = 0-387-98585-9}} p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7*
 
{{Honeycombs}}
 
[[Category:8-polytopes]]

Revision as of 14:41, 10 January 2014

331 honeycomb
(no image)
Type Uniform tessellation
Schläfli symbol {3,3,3,33,1}
Coxeter symbol 331
Coxeter-Dynkin diagram Template:CDD
7-face types 321
{36}
6-face types 221
{35}
5-face types 211
{34}
4-face type {33}
Cell type {32}
Face type {3}
Face figure 031
Edge figure 131
Vertex figure 231
Coxeter group E~7, [33,3,1]
Properties vertex-transitive

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schlafli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

Template:CDD

Removing the node on the short branch leaves the 6-simplex facet:

Template:CDD

Removing the node on the end of the 3-length branch leaves the 321 facet:

Template:CDD

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 231 polytope.

Template:CDD

The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (131).

Template:CDD

The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (031).

Template:CDD

The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes tetrahedral prism {}×{3,3}.

Template:CDD

Kissing number

Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions; its kissing number is 126, represented by the vertices of its vertex figure 231.

E7 lattice

E~7 contains A~7 as a subgroup of index 144.[1] Both E~7 and A~7 can be seen as affine extension from A7 from different nodes:

The vertex arrangement of 331 is called the E7 lattice.[2] The E7 lattice can also be expressed as a union of the vertices of two A7 lattices, also called A72:

Template:CDD = Template:CDD + Template:CDD

The E7* lattice (also called E72)[3] has double the symmetry, represented by [[3,33,3]]. The Voronoi cell of the E7* lattice is the 132 polytope, and voronoi tessellation the 133 honeycomb.[4] The E7* lattice is constructed by 2 copies of the E7 lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A7* lattices, also called A74:

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

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron. Tucao this car was bought when two places is not satisfied, a no seat ventilation, a panoramic sunroof no, not France optional.
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See also

References

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  • H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
  • 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] GoogleBook
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
  • R. T. Worley, The Voronoi Region of E7*. SIAM J. Disc. Math., 1.1 (1988), 134-141.
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    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7*

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  1. ↑ N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p 177
  2. ↑ http://www2.research.att.com/~njas/lattices/E7.html
  3. ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
  4. ↑ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
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