# Rectified 6-simplexes

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{| class="wikitable skin-invert-image" style="float:right; margin-left:1em; width:450px"
|- align=center valign=top
|150px<BR>[6-simplex](/source/6-simplex)<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
|150px<BR>Rectified 6-simplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node}}
|150px<BR>Birectified 6-simplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node}}
|-
!colspan=3|[Orthogonal projection](/source/Orthogonal_projection)s in A<sub>6</sub> [Coxeter plane](/source/Coxeter_plane)
|}
In six-dimensional [geometry](/source/geometry), a '''rectified 6-simplex''' is a convex [uniform 6-polytope](/source/uniform_6-polytope), being a [rectification](/source/Rectification_(geometry)) of the regular [6-simplex](/source/6-simplex).

There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the ''rectified 6-simplex'' are located at the edge-centers of the ''6-simplex''. Vertices of the ''birectified 6-simplex'' are located in the triangular face centers of the ''6-simplex''.
{{-}}

== Rectified 6-simplex ==
{{Uniform polypeton db|Uniform polypeton stat table|ril}}
[E. L. Elte](/source/Emanuel_Lodewijk_Elte) identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|6}}. It is also called '''0<sub>4,1</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea}}.

=== Alternate names ===
* Rectified heptapeton (Acronym: ril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/ril.htm (o3x3o3o3o3o - ril)]}}

=== Coordinates ===
The vertices of the ''rectified 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on [facets](/source/Facet_(geometry)) of the [rectified 7-orthoplex](/source/rectified_7-orthoplex).

=== Images ===
{{6-simplex Coxeter plane graphs|t1|150}}

== Birectified 6-simplex ==
{| class="wikitable" style="float:right; margin-left:10px; width:250px"
!style="background:#e7dcc3" colspan=2|Birectified 6-simplex
|-
|style=background:#e7dcc3|Type||[uniform 6-polytope](/source/uniform_6-polytope)
|-
|style=background:#e7dcc3|Class||[A6 polytope](/source/A6_polytope)
|-
|style=background:#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)|| t<sub>2</sub>{3,3,3,3,3}<BR>2r{3<sup>5</sup>} = {3<sup>3,2</sup>}<BR>or <math>\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\}</math>
|-
|style=background:#e7dcc3|[Coxeter symbol](/source/Coxeter_symbol)|| 0<sub>32</sub>
|-
|style=background:#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)s||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node}}<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea}}
|-
|style=background:#e7dcc3|5-faces||14 total:<BR>7 [t<sub>1</sub>{3,3,3,3}](/source/Rectified_5-simplex)<BR>7 [t<sub>2</sub>{3,3,3,3}](/source/Birectified_5-simplex)
|-
|style=background:#e7dcc3|4-faces||84
|-
|style=background:#e7dcc3|Cells||245
|-
|style=background:#e7dcc3|Faces||350
|-
|style=background:#e7dcc3|Edges||210
|-
|style=background:#e7dcc3|Vertices||35
|-
|style=background:#e7dcc3|[Vertex figure](/source/Vertex_figure)||{3}x{3,3}
|-
|style=background:#e7dcc3|[Petrie polygon](/source/Petrie_polygon)||[Heptagon](/source/Heptagon)
|-
|style=background:#e7dcc3|[Coxeter group](/source/Coxeter_group)s||A<sub>6</sub>, [3,3,3,3,3]
|-
|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|6}}. It is also called '''0<sub>3,2</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea}}.

=== Alternate names ===
* Birectified heptapeton (Acronym: bril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/bril.htm (o3o3x3o3o3o - bril)]}}

=== Coordinates ===
The vertices of the ''birectified 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on [facets](/source/Facet_(geometry)) of the [birectified 7-orthoplex](/source/birectified_7-orthoplex).

=== Images ===
{{6-simplex Coxeter plane graphs|t2|150}}

== Related uniform 6-polytopes ==
The rectified 6-simplex polytope is the [vertex figure](/source/vertex_figure) of the [7-demicube](/source/7-demicube), and the [edge figure](/source/edge_figure) of the uniform [2<sub>41</sub> polytope](/source/2_41_polytope).

These polytopes are a part of 35 [uniform 6-polytopes](/source/Uniform_6-polytope) based on the [3,3,3,3,3] [Coxeter group](/source/Coxeter_group), all shown here in A<sub>6</sub> [Coxeter plane](/source/Coxeter_plane) [orthographic projection](/source/orthographic_projection)s.

{{Heptapeton family}}

== 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|polypeta.htm|6D uniform polytopes (polypeta) with acronyms}} o3x3o3o3o3o - ril, o3o3x3o3o3o - bril {{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:6-polytopes

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Adapted from the Wikipedia article [Rectified 6-simplexes](https://en.wikipedia.org/wiki/Rectified_6-simplexes) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rectified_6-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.