View source for Rectified 8-simplexes

{| class=wikitable align=right style="margin-left:1em;"
|- align=center valign=top
|[[File:8-simplex t0.svg|150px]]<BR>8-simplex
|[[File:8-simplex t1.svg|150px]]<BR>Rectified 8-simplex
|- align=center valign=top
|[[File:8-simplex t2.svg|150px]]<BR>Birectified 8-simplex
|[[File:8-simplex t3.svg|150px]]<BR>Trirectified 8-simplex
|-
!colspan=4|[[Orthogonal projection]]s in A<sub>8</sub> [[Coxeter plane]]
|}
In eight-dimensional [[geometry]], a '''rectified 8-simplex''' is a convex [[uniform 8-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[8-simplex]].

There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the [[tetrahedron|tetrahedral]] cell centers of the 8-simplex. 
{{clear}}

== Rectified 8-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Rectified 8-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>61</sub>
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| t<sub>1</sub>{3<sup>7</sup>}<BR>r{3<sup>7</sup>} = {3<sup>6,1</sup>}<BR>or <math>\left\{\begin{array}{l}3, 3, 3, 3, 3,3\\3\end{array}\right\}</math>
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>or {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|-
| style="background:#e7dcc3;"|7-faces||18
|-
| style="background:#e7dcc3;"|6-faces||108
|-
| style="background:#e7dcc3;"|5-faces||336
|-
| style="background:#e7dcc3;"|4-faces||630
|-
| style="background:#e7dcc3;"|Cells||756
|-
| style="background:#e7dcc3;"|Faces||588
|-
| style="background:#e7dcc3;"|Edges||252
|-
| style="background:#e7dcc3;"|Vertices||36
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||7-simplex prism, {}×{3,3,3,3,3}
|-
| style="background:#e7dcc3;"|[[Petrie polygon]]||[[enneagon]]
|-
| style="background:#e7dcc3;"|[[Coxeter group]]||A<sub>8</sub>, [3<sup>7</sup>], order 362880
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}
[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|8}}. It is also called '''0<sub>6,1</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.
=== Coordinates ===

The [[Cartesian coordinate]]s of the vertices of the ''rectified 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on [[Facet (geometry)|facets]] of the [[rectified 9-orthoplex]].

=== Images ===
{{8-simplex Coxeter plane graphs|t1|120}}

{{-}}
== Birectified 8-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
! style="background:#e7dcc3;" colspan="2"|Birectified 8-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>52</sub>
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| t<sub>2</sub>{3<sup>7</sup>}<BR>2r{3<sup>7</sup>} = {3<sup>5,2</sup>} or<BR><math>\left\{\begin{array}{l}3, 3, 3, 3, 3\\3, 3\end{array}\right\}</math>
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}<BR>or {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea|3a|nodea}}
|-
| style="background:#e7dcc3;"|7-faces||18
|-
| style="background:#e7dcc3;"|6-faces||144
|-
| style="background:#e7dcc3;"|5-faces||588
|-
| style="background:#e7dcc3;"|4-faces||1386
|-
| style="background:#e7dcc3;"|Cells||2016
|-
| style="background:#e7dcc3;"|Faces||1764
|-
| style="background:#e7dcc3;"|Edges||756
|-
| style="background:#e7dcc3;"|Vertices||84
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||{3}×{3,3,3,3}
|-
| style="background:#e7dcc3;"|[[Coxeter group]]||A<sub>8</sub>, [3<sup>7</sup>], order 362880
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}
[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|2|8}}. It is also called '''0<sub>5,2</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea|3a|nodea}}.

The ''birectified 8-simplex'' is the [[vertex figure]] of the [[1 52 honeycomb|1<sub>52</sub> honeycomb]].
=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of the ''birectified 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on [[Facet (geometry)|facets]] of the [[birectified 9-orthoplex]].

=== Images ===
{{8-simplex Coxeter plane graphs|t2|120}}

== Trirectified 8-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
! style="background:#e7dcc3;" colspan="2"|Trirectified 8-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>43</sub>
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| t<sub>3</sub>{3<sup>7</sup>}<BR>3r{3<sup>7</sup>} = {3<sup>4,3</sup>} or<BR><math>\left\{\begin{array}{l}3, 3, 3, 3\\3, 3,3 \end{array}\right\}</math>
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}<BR>or {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3a|nodea}}
|-
| style="background:#e7dcc3;"|7-faces||9 + 9
|-
| style="background:#e7dcc3;"|6-faces|| 36 + 72 + 36
|-
| style="background:#e7dcc3;"|5-faces|| 84 + 252 + 252 + 84
|-
| style="background:#e7dcc3;"|4-faces|| 126 + 504 + 756 + 504
|-
| style="background:#e7dcc3;"|Cells|| 630 + 1260 + 1260
|-
| style="background:#e7dcc3;"|Faces|| 1260 + 1680
|-
| style="background:#e7dcc3;"|Edges||1260
|-
| style="background:#e7dcc3;"|Vertices||126
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||{3,3}×{3,3,3}
|-
| style="background:#e7dcc3;"|[[Petrie polygon]]||[[enneagon]]
|-
| style="background:#e7dcc3;"|[[Coxeter group]]||A<sub>7</sub>, [3<sup>7</sup>], order 362880
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}
[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|3|8}}. It is also called '''0<sub>4,3</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3a|nodea}}.
=== Coordinates ===
The [[Cartesian coordinate]]s of the vertices of the ''trirectified 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on [[Facet (geometry)|facets]] of the [[trirectified 9-orthoplex]].

=== Images ===
{{8-simplex Coxeter plane graphs|t3|120}}

== Related polytopes ==

This polytope is the [[vertex figure]] of the [[9-demicube]], and the edge figure of the uniform [[2 61 honeycomb|2<sub>61</sub> honeycomb]].

It is also one of 135 [[8-polytope#The A8 .5B3,3,3,3,3,3,3.5D family (8-simplex)|uniform 8-polytopes]] with A<sub>8</sub> symmetry.
{{Enneazetton family}}

== Notes ==
{{reflist}}

== References==
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 
** '''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]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 
* {{KlitzingPolytopes|polyzetta.htm|8D|Uniform polytopes (polyzetta)}} o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene

== External links ==
* [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}

[[Category:8-polytopes]]