{| class="wikitable skin-invert-image" style="float:right; margin-left:1em; width:450px" |- align=center valign=top |150px<BR>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 projections in A<sub>6</sub> Coxeter plane |} In six-dimensional geometry, a '''rectified 6-simplex''' is a convex uniform 6-polytope, being a rectification of the regular 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 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 of the 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 |- |style=background:#e7dcc3|Class||A6 polytope |- |style=background:#e7dcc3|Schläfli 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|| 0<sub>32</sub> |- |style=background:#e7dcc3|Coxeter diagrams||{{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}<BR>7 t<sub>2</sub>{3,3,3,3} |- |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||{3}x{3,3} |- |style=background:#e7dcc3|Petrie polygon||Heptagon |- |style=background:#e7dcc3|Coxeter groups||A<sub>6</sub>, [3,3,3,3,3] |- |style=background:#e7dcc3|Properties||convex |} E. L. 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 of the birectified 7-orthoplex.
=== Images === {{6-simplex Coxeter plane graphs|t2|150}}
== Related uniform 6-polytopes == The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 2<sub>41</sub> polytope.
These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A<sub>6</sub> Coxeter plane orthographic projections.
{{Heptapeton family}}
== Notes == {{reflist}}
== References == * 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 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 ''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