View source for Rectified 7-simplexes

{{Short description|Convex uniform 7-polytope in seven-dimensional geometry}}
{| class=wikitable align=right width=300
|- align=center valign=top
|[[File:7-simplex t0.svg|150px]]<BR>7-simplex<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
|[[File:7-simplex t1.svg|150px]]<BR>Rectified 7-simplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
|- align=center valign=top
|[[File:7-simplex t2.svg|150px]]<BR>Birectified 7-simplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:7-simplex t3.svg|150px]]<BR>Trirectified 7-simplex<BR>{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}
|-
!colspan=3|[[Orthogonal projection]]s in A<sub>7</sub> [[Coxeter plane]]
|}

In seven-dimensional [[geometry]], a '''rectified 7-simplex''' is a convex [[uniform 7-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[7-simplex]].

There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the ''rectified 7-simplex'' are located at the edge-centers of the ''7-simplex''. Vertices of the ''birectified 7-simplex'' are located in the triangular face centers of the ''7-simplex''. Vertices of the ''trirectified 7-simplex'' are located in the [[tetrahedron|tetrahedral]] cell centers of the ''7-simplex''.

== Rectified 7-simplex ==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Rectified 7-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 7-polytope]]
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>51</sub>
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| r{3<sup>6</sup>} = {3<sup>5,1</sup>}<BR>or <math>\left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|3|node_1|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}}
|-
|bgcolor=#e7dcc3|6-faces||16
|-
|bgcolor=#e7dcc3|5-faces||84
|-
|bgcolor=#e7dcc3|4-faces||224
|-
|bgcolor=#e7dcc3|Cells||350
|-
|bgcolor=#e7dcc3|Faces||336
|-
|bgcolor=#e7dcc3|Edges||168
|-
|bgcolor=#e7dcc3|Vertices||28
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||6-simplex prism
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Octagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||A<sub>7</sub>, [3<sup>6</sup>], order 40320
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The rectified 7-simplex is the [[edge figure]] of the [[2 51 honeycomb|2<sub>51</sub> honeycomb]]. It is called '''0<sub>5,1</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|7}}.

=== Alternate names===
* Rectified octaexon (Acronym: roc) (Jonathan Bowers)

=== Coordinates ===

The vertices of the ''rectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on [[Facet (geometry)|facets]] of the [[rectified 8-orthoplex]].

=== Images ===

{{7-simplex Coxeter plane graphs|t1|150}}
{{-}}

== Birectified 7-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Birectified 7-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 7-polytope]]
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>42</sub>
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2r{3,3,3,3,3,3} = {3<sup>4,2</sup>}<BR>or <math>\left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}<BR>Or {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|6-faces||16:<BR>8 [[Rectified 6-simplex|r{3<sup>5</sup>}]] [[File:6-simplex t1.svg|25px]]<BR>8 [[Birectified 6-simplex|2r{3<sup>5</sup>}]] [[File:6-simplex t2.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||112:<BR>28 [[5-simplex|{3<sup>4</sup>}]] [[File:5-simplex t0.svg|25px]]<BR>56 [[Rectified 5-simplex|r{3<sup>4</sup>}]] [[File:Rectified 5-simplex.png|25px]]<BR>28 [[Birectified 5-simplex|2r{3<sup>4</sup>}]] [[File:5-simplex t2.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||392:<BR>168 [[5-cell|{3<sup>3</sup>}]] [[File:4-simplex t0.svg|25px]]<BR>(56+168) [[Rectified 5-cell|r{3<sup>3</sup>}]] [[File:5-simplex t1.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||770:<BR>(420+70) [[Tetrahedron|{3,3}]] [[File:3-simplex t0.svg|25px]]<BR>280 [[Octahedron|{3,4}]] [[File:3-simplex t1.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||840:<BR>(280+560) [[Triangle|{3}]]
|-
|bgcolor=#e7dcc3|Edges||420
|-
|bgcolor=#e7dcc3|Vertices||56
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{3}x{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||A<sub>7</sub>, [3<sup>6</sup>], order 40320
|-
|bgcolor=#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|7}}. It is also called '''0<sub>4,2</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea}}.

=== Alternate names===
* Birectified octaexon (Acronym: broc) (Jonathan Bowers)

=== Coordinates ===

The vertices of the ''birectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on [[Facet (geometry)|facets]] of the [[birectified 8-orthoplex]].

=== Images ===

{{7-simplex Coxeter plane graphs|t2|150}}
{{-}}

== Trirectified 7-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Trirectified 7-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 7-polytope]]
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>33</sub>
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 3r{3<sup>6</sup>} = {3<sup>3,3</sup>}<BR>or <math>\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}<BR>Or {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes}}
|-
|bgcolor=#e7dcc3|6-faces||16 [[Birectified 6-simplex|2r{3<sup>5</sup>}]]
|-
|bgcolor=#e7dcc3|5-faces||112
|-
|bgcolor=#e7dcc3|4-faces||448
|-
|bgcolor=#e7dcc3|Cells||980
|-
|bgcolor=#e7dcc3|Faces||1120
|-
|bgcolor=#e7dcc3|Edges||560
|-
|bgcolor=#e7dcc3|Vertices||70
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{3,3}x{3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||A<sub>7</sub>×2, <nowiki>[[</nowiki>3<sup>6</sup>]], order 80640
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Facet-transitive|isotopic]]
|}
The ''trirectified 7-simplex'' is the [[intersection (set theory)|intersection]] of two regular [[7-simplex]]es in [[dual polytope|dual]] configuration.

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|3|7}}.

This polytope is the [[vertex figure]] of the [[1 33 honeycomb|1<sub>33</sub> honeycomb]]. It is called '''0<sub>3,3</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes}}.

=== Alternate names===
* Hexadecaexon (Acronym: he) (Jonathan Bowers)

=== Coordinates ===

The vertices of the ''trirectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on [[Facet (geometry)|facets]] of the [[trirectified 8-orthoplex]].

The ''trirectified 7-simplex'' is the [[intersection (set theory)|intersection]] of two regular 7-simplices in [[dual polytope|dual]] configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).

=== Images ===
{{7-simplex2 Coxeter plane graphs|t3|150}}

=== Related polytopes===
{{Isotopic uniform simplex polytopes}}

== Related polytopes ==

These polytopes are three of 71 [[uniform 7-polytope]]s with A<sub>7</sub> symmetry.
{{Octaexon family}}

==See also==
*[[List of A7 polytopes]]

== 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|polyexa.htm|7D|uniform polytopes (polyexa)}} o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he

== 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:7-polytopes]]