Rectified 7-simplexes

{{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<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 projections in A₇ Coxeter plane |}

In seven-dimensional geometry, a '''rectified 7-simplex''' is a convex uniform 7-polytope, being a 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 tetrahedral 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 |- |style="background:#e7dcc3"|Coxeter symbol|| 0₅₁ |- |style="background:#e7dcc3"|Schläfli symbol|| r{3⁶} = {3^5,1}<BR>or <math>\left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\}</math> |- |style="background:#e7dcc3"|Coxeter diagrams||{{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||6-simplex prism |- |style="background:#e7dcc3"|Petrie polygon||Octagon |- |style="background:#e7dcc3"|Coxeter group||A₇, [3⁶], order 40320 |- |style="background:#e7dcc3"|Properties||convex |} The rectified 7-simplex is the edge figure of the 2₅₁ honeycomb. It is called '''0_5,1''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

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){{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 of the 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 |- |style="background:#e7dcc3"|Coxeter symbol|| 0₄₂ |- |style="background:#e7dcc3"|Schläfli symbol|| 2r{3,3,3,3,3,3} = {3^4,2}<BR>or <math>\left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\}</math> |- |style="background:#e7dcc3"|Coxeter diagrams||{{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{3⁵} 25px|class=skin-invert<BR>8 2r{3⁵} 25px|class=skin-invert |- |style="background:#e7dcc3"|5-faces||112:<BR>28 {3⁴} 25px|class=skin-invert<BR>56 r{3⁴} 25px|class=skin-invert<BR>28 2r{3⁴} 25px|class=skin-invert |- |style="background:#e7dcc3"|4-faces||392:<BR>168 {3³} 25px|class=skin-invert<BR>(56+168) r{3³} 25px|class=skin-invert |- |style="background:#e7dcc3"|Cells||770:<BR>(420+70) {3,3} 25px|class=skin-invert<BR>280 {3,4} 25px|class=skin-invert |- |style="background:#e7dcc3"|Faces||840:<BR>(280+560) {3} |- |style="background:#e7dcc3"|Edges||420 |- |style="background:#e7dcc3"|Vertices||56 |- |style="background:#e7dcc3"|Vertex figure||{3}x{3,3,3} |- |style="background:#e7dcc3"|Coxeter group||A₇, [3⁶], order 40320 |- |style="background:#e7dcc3"|Properties||convex |} E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|2|7}}. It is also called '''0_4,2''' 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 of the 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 |- |style="background:#e7dcc3"|Coxeter symbol|| 0₃₃ |- |style="background:#e7dcc3"|Schläfli symbol|| 3r{3⁶} = {3^3,3}<BR>or <math>\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}</math> |- |style="background:#e7dcc3"|Coxeter diagrams||{{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{3⁵} |- |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||{3,3}x{3,3} |- |style="background:#e7dcc3"|Coxeter group||A₇×2, <nowiki></nowiki>3⁶, order 80640 |- |style="background:#e7dcc3"|Properties||convex, isotopic |} The ''trirectified 7-simplex'' is the intersection of two regular 7-simplexes in dual configuration.

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 1₃₃ honeycomb. It is called '''0_3,3''' 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 of the trirectified 8-orthoplex.

The ''trirectified 7-simplex'' is the intersection of two regular 7-simplices in 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|120}}

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

== Related polytopes == These polytopes are three of 71 uniform 7-polytopes with A₇ symmetry. {{Octaexon family}}

== See also == *List of A7 polytopes

== 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|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