# Rectified 7-simplexes

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Rectified_7-simplexes
> Markdown URL: https://mediated.wiki/source/Rectified_7-simplexes.md
> Source: https://en.wikipedia.org/wiki/Rectified_7-simplexes
> Source revision: 1355614410
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Convex uniform 7-polytope in seven-dimensional geometry}}
{| class="wikitable skin-invert-image" style="float:right; margin-left:8px; width:300px"
|- align=center valign=top
|150px<BR>[7-simplex](/source/7-simplex)<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
|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
|150px<BR>Birectified 7-simplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|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](/source/Orthogonal_projection)s in A<sub>7</sub> [Coxeter plane](/source/Coxeter_plane)
|}

In seven-dimensional [geometry](/source/geometry), a '''rectified 7-simplex''' is a convex [uniform 7-polytope](/source/uniform_7-polytope), being a [rectification](/source/Rectification_(geometry)) of the regular [7-simplex](/source/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 [tetrahedral](/source/tetrahedron) cell centers of the ''7-simplex''.
{{-}}

== Rectified 7-simplex ==
{| class="wikitable" style="float:right; margin-left:10px; width:250px"
!style="background:#e7dcc3" colspan=2|Rectified 7-simplex
|-
|style="background:#e7dcc3"|Type||[uniform 7-polytope](/source/uniform_7-polytope)
|-
|style="background:#e7dcc3"|[Coxeter symbol](/source/Coxeter_symbol)|| 0<sub>51</sub>
|-
|style="background:#e7dcc3"|[Schläfli symbol](/source/Schl%C3%A4fli_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>
|-
|style="background:#e7dcc3"|[Coxeter diagram](/source/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}}
|-
|style="background:#e7dcc3"|6-faces||16
|-
|style="background:#e7dcc3"|5-faces||84
|-
|style="background:#e7dcc3"|4-faces||224
|-
|style="background:#e7dcc3"|Cells||350
|-
|style="background:#e7dcc3"|Faces||336
|-
|style="background:#e7dcc3"|Edges||168
|-
|style="background:#e7dcc3"|Vertices||28
|-
|style="background:#e7dcc3"|[Vertex figure](/source/Vertex_figure)||6-simplex prism
|-
|style="background:#e7dcc3"|[Petrie polygon](/source/Petrie_polygon)||[Octagon](/source/Octagon)
|-
|style="background:#e7dcc3"|[Coxeter group](/source/Coxeter_group)||A<sub>7</sub>, [3<sup>6</sup>], order 40320
|-
|style="background:#e7dcc3"|Properties||[convex](/source/Convex_polytope)
|}
The rectified 7-simplex is the [edge figure](/source/edge_figure) of the [2<sub>51</sub> honeycomb](/source/2_51_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}}.

[E. L. Elte](/source/Emanuel_Lodewijk_Elte) identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|7}}.

=== Alternate names ===
* Rectified octaexon (Acronym: roc) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/roc.htm (o3x3o3o3o3o3o - roc)]}}

=== 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 [facets](/source/Facet_(geometry)) of the [rectified 8-orthoplex](/source/rectified_8-orthoplex).

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

== Birectified 7-simplex ==
{| class="wikitable" style="float:right; margin-left:10px; width:280px"
!style="background:#e7dcc3" colspan=2|Birectified 7-simplex
|-
|style="background:#e7dcc3"|Type||[uniform 7-polytope](/source/uniform_7-polytope)
|-
|style="background:#e7dcc3"|[Coxeter symbol](/source/Coxeter_symbol)|| 0<sub>42</sub>
|-
|style="background:#e7dcc3"|[Schläfli symbol](/source/Schl%C3%A4fli_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>
|-
|style="background:#e7dcc3"|[Coxeter diagram](/source/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}}
|-
|style="background:#e7dcc3"|6-faces||16:<BR>8 [r{3<sup>5</sup>}](/source/Rectified_6-simplex) 25px|class=skin-invert<BR>8 [2r{3<sup>5</sup>}](/source/Birectified_6-simplex) 25px|class=skin-invert
|-
|style="background:#e7dcc3"|5-faces||112:<BR>28 [{3<sup>4</sup>}](/source/5-simplex) 25px|class=skin-invert<BR>56 [r{3<sup>4</sup>}](/source/Rectified_5-simplex) 25px|class=skin-invert<BR>28 [2r{3<sup>4</sup>}](/source/Birectified_5-simplex) 25px|class=skin-invert
|-
|style="background:#e7dcc3"|4-faces||392:<BR>168 [{3<sup>3</sup>}](/source/5-cell) 25px|class=skin-invert<BR>(56+168) [r{3<sup>3</sup>}](/source/Rectified_5-cell) 25px|class=skin-invert
|-
|style="background:#e7dcc3"|Cells||770:<BR>(420+70) [{3,3}](/source/Tetrahedron) 25px|class=skin-invert<BR>280 [{3,4}](/source/Octahedron) 25px|class=skin-invert
|-
|style="background:#e7dcc3"|Faces||840:<BR>(280+560) [{3}](/source/Triangle)
|-
|style="background:#e7dcc3"|Edges||420
|-
|style="background:#e7dcc3"|Vertices||56
|-
|style="background:#e7dcc3"|[Vertex figure](/source/Vertex_figure)||{3}x{3,3,3}
|-
|style="background:#e7dcc3"|[Coxeter group](/source/Coxeter_group)||A<sub>7</sub>, [3<sup>6</sup>], order 40320
|-
|style="background:#e7dcc3"|Properties||[convex](/source/Convex_polytope)
|}
[E. L. Elte](/source/Emanuel_Lodewijk_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){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/broc.htm (o3o3x3o3o3o3o - broc)]}}

=== 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 [facets](/source/Facet_(geometry)) of the [birectified 8-orthoplex](/source/birectified_8-orthoplex).

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

== Trirectified 7-simplex ==
{| class="wikitable" style="float:right; margin-left:10px; width:280px"
!style="background:#e7dcc3" colspan=2|Trirectified 7-simplex
|-
|style="background:#e7dcc3"|Type||[uniform 7-polytope](/source/uniform_7-polytope)
|-
|style="background:#e7dcc3"|[Coxeter symbol](/source/Coxeter_symbol)|| 0<sub>33</sub>
|-
|style="background:#e7dcc3"|[Schläfli symbol](/source/Schl%C3%A4fli_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>
|-
|style="background:#e7dcc3"|[Coxeter diagram](/source/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}}
|-
|style="background:#e7dcc3"|6-faces||16 [2r{3<sup>5</sup>}](/source/Birectified_6-simplex)
|-
|style="background:#e7dcc3"|5-faces||112
|-
|style="background:#e7dcc3"|4-faces||448
|-
|style="background:#e7dcc3"|Cells||980
|-
|style="background:#e7dcc3"|Faces||1120
|-
|style="background:#e7dcc3"|Edges||560
|-
|style="background:#e7dcc3"|Vertices||70
|-
|style="background:#e7dcc3"|[Vertex figure](/source/Vertex_figure)||{3,3}x{3,3}
|-
|style="background:#e7dcc3"|[Coxeter group](/source/Coxeter_group)||A<sub>7</sub>×2, <nowiki>[</nowiki>3<sup>6</sup>](/source/%3C%2Fnowiki%3E3%3Csup%3E6%3C%2Fsup%3E), order 80640
|-
|style="background:#e7dcc3"|Properties||[convex](/source/Convex_polytope), [isotopic](/source/Facet-transitive)
|}
The ''trirectified 7-simplex'' is the [intersection](/source/intersection_(set_theory)) of two regular [7-simplex](/source/7-simplex)es in [dual](/source/dual_polytope) configuration.

[E. L. Elte](/source/Emanuel_Lodewijk_Elte) identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|3|7}}.

This polytope is the [vertex figure](/source/vertex_figure) of the [1<sub>33</sub> honeycomb](/source/1_33_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){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/he.htm (o3o3o3x3o3o3o - he)]}}

=== 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 [facets](/source/Facet_(geometry)) of the [trirectified 8-orthoplex](/source/trirectified_8-orthoplex).

The ''trirectified 7-simplex'' is the [intersection](/source/intersection_(set_theory)) of two regular 7-simplices in [dual](/source/dual_polytope) 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|120}}

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

== Related polytopes ==
These polytopes are three of 71 [uniform 7-polytope](/source/uniform_7-polytope)s with A<sub>7</sub> symmetry.
{{Octaexon family}}

== See also ==
*[List of A7 polytopes](/source/List_of_A7_polytopes)

== Notes ==
{{reflist}}

== References ==
* [H.S.M. Coxeter](/source/Harold_Scott_MacDonald_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 Ivić Weiss, Wiley-Interscience Publication, 1995, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}
*** (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](/source/Norman_Johnson_(mathematician)) ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 
* {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa) with acronyms}} o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - broc, o3o3o3x3o3o3o - he {{sfn whitelist| CITEREFKlitzing}}

== 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

---
Adapted from the Wikipedia article [Rectified 7-simplexes](https://en.wikipedia.org/wiki/Rectified_7-simplexes) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rectified_7-simplexes?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.