Timeline of mathematics: Difference between revisions

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{{DISPLAYTITLE:2<sub><span style="display:none"> </span>51</sub> honeycomb}}
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{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|2<sub>51</sub> honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||[[9-polytope#Regular_and_uniform_honeycombs|Uniform tessellation]]
|-
|bgcolor=#e7dcc3|Family||[[Uniform 2 k1 polytope|2<sub>k1</sub> polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3<sup>5,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| 2<sub>51</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|8-face types||'''[[2 41 polytope|2<sub>41</sub>]]'''[[File:Gosset 2 41 petrie.svg|25px]]<BR>[[8-simplex|{3<sup>7</sup>}]][[Image:8-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|7-face types||'''[[2 31 polytope|2<sub>31</sub>]]'''[[Image:Gosset 2 31 polytope.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|Cells||[[tetrahedron|{3<sup>2</sup>}]][[Image:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle|{3}]][[Image:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Edge figure]]||'''[[rectified 6-simplex|0<sub>51</sub>]]''' [[File:6-simplex t1.svg|30px]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||'''[[8-demicube|1<sub>51</sub>]]''' [[File:8-demicube.svg|30px]]
|-
|bgcolor=#e7dcc3|[[Edge figure]]||'''[[Rectified 7-simplex|0<sub>51</sub>]]''' [[File:7-simplex t1.svg|30px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{E}}_8</math>, [3<sup>5,2,1</sup>]
|}
In 8-dimensional [[geometry]], the '''2<sub>51</sub>''' honeycomb is a space-filling [[uniform tessellation]]. It is composed of [[Gosset 2 41 polytope|2<sub>41</sub> polytope]] and [[8-simplex]] [[Facet (geometry)|facets]] arranged in a [[8-demicube]] [[vertex figure]]. It is the final figure in the [[uniform 2 k1 polytope|2<sub>k1</sub> family]].


==Construction==
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It is created by a [[Wythoff construction]] upon a set of 9 [[hyperplane]] mirrors in 8-dimensional space.
 
The facet information can be extracted from its [[Coxeter-Dynkin diagram]].
: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
 
Removing the node on the short branch leaves the [[8-simplex]].  
: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
 
Removing the node on the end of the 5-length branch leaves the [[Gosset 2 41 polytope|2<sub>41</sub>]].  
: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
 
The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes the [[8-demicube]], 1<sub>51</sub>.
: {{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
 
The [[edge figure]] is the vertex figure of the vertex figure. This makes the [[rectified 7-simplex]], 0<sub>51</sub>.
: {{CDD|branch_10|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
 
{{-}}
 
== Related polytopes and honeycombs==
{{2 k1 polytopes}}
 
== See also ==
* [[5 21 honeycomb|5<sub>21</sub> honeycomb]]
* [[1 52 honeycomb|1<sub>52</sub> honeycomb]]
 
==References==
* [[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)
* [[Harold Scott MacDonald Coxeter|Coxeter]] ''Regular Polytopes'' (1963), Macmillian Company
** ''Regular Polytopes'', Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope)
* '''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]
 
{{Honeycombs}}
 
[[Category:9-polytopes]]
 
 
{{geometry-stub}}

Latest revision as of 11:53, 9 January 2015

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