Rectified 7-simplexes#Trirectified 7-simplex

{{Short description|Convex uniform 7-polytope in seven-dimensional geometry}}

{{No footnotes|date=April 2025}}

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|150px
7-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}

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Rectified 7-simplex
{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}

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Birectified 7-simplex
{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}

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Trirectified 7-simplex
{{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.

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Rectified 7-simplex

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!bgcolor=#e7dcc3 colspan=2|Rectified 7-simplex

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Coxeter symbol051
bgcolor=#e7dcc3|Schläfli symbolr{36} = {35,1}
or \left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
Or {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
bgcolor=#e7dcc3|6-faces16
bgcolor=#e7dcc3|5-faces84
bgcolor=#e7dcc3|4-faces224
bgcolor=#e7dcc3|Cells350
bgcolor=#e7dcc3|Faces336
bgcolor=#e7dcc3|Edges168
bgcolor=#e7dcc3|Vertices28
bgcolor=#e7dcc3|Vertex figure6-simplex prism
bgcolor=#e7dcc3|Petrie polygonOctagon
bgcolor=#e7dcc3|Coxeter groupA7, [36], order 40320
bgcolor=#e7dcc3|Propertiesconvex

The rectified 7-simplex is the edge figure of the 251 honeycomb. It is called 05,1 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)

= 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|150}}

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Birectified 7-simplex

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!bgcolor=#e7dcc3 colspan=2|Birectified 7-simplex

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Coxeter symbol042
bgcolor=#e7dcc3|Schläfli symbol2r{3,3,3,3,3,3} = {34,2}
or \left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
Or {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea}}
bgcolor=#e7dcc3|6-faces16:
8 r{35} 25px
8 2r{35} 25px
bgcolor=#e7dcc3|5-faces112:
28 {34} 25px
56 r{34} 25px
28 2r{34} 25px
bgcolor=#e7dcc3|4-faces392:
168 {33} 25px
(56+168) r{33} 25px
bgcolor=#e7dcc3|Cells770:
(420+70) {3,3} 25px
280 {3,4} 25px
bgcolor=#e7dcc3|Faces840:
(280+560) {3}
bgcolor=#e7dcc3|Edges420
bgcolor=#e7dcc3|Vertices56
bgcolor=#e7dcc3|Vertex figure{3}x{3,3,3}
bgcolor=#e7dcc3|Coxeter groupA7, [36], order 40320
bgcolor=#e7dcc3|Propertiesconvex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|2|7}}. It is also called 04,2 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)

= 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|150}}

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Trirectified 7-simplex

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!bgcolor=#e7dcc3 colspan=2|Trirectified 7-simplex

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Coxeter symbol033
bgcolor=#e7dcc3|Schläfli symbol3r{36} = {33,3}
or \left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}
Or {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes}}
bgcolor=#e7dcc3|6-faces16 2r{35}
bgcolor=#e7dcc3|5-faces112
bgcolor=#e7dcc3|4-faces448
bgcolor=#e7dcc3|Cells980
bgcolor=#e7dcc3|Faces1120
bgcolor=#e7dcc3|Edges560
bgcolor=#e7dcc3|Vertices70
bgcolor=#e7dcc3|Vertex figure{3,3}x{3,3}
bgcolor=#e7dcc3|Coxeter groupA7×2, 36, order 80640
bgcolor=#e7dcc3|Propertiesconvex, 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 03,3 for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes}}.

= Alternate names =

  • Hexadecaexon (Acronym: he) (Jonathan Bowers)

= 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|150}}

= Related polytopes =

{{Isotopic uniform simplex polytopes}}

Related polytopes

These polytopes are three of 71 uniform 7-polytopes with A7 symmetry.

{{Octaexon family}}

See also

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 Ivic 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)}} o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he