Rectified 7-simplexes#Trirectified 7-simplex

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}}.

• Rectified octaexon (Acronym: roc) (Jonathan Bowers)

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.

{{7-simplex Coxeter plane graphs|t1|150}}

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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}}.

• Birectified octaexon (Acronym: broc) (Jonathan Bowers)

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.

{{7-simplex Coxeter plane graphs|t2|150}}

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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}}.

• Hexadecaexon (Acronym: he) (Jonathan Bowers)

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).

{{7-simplex2 Coxeter plane graphs|t3|150}}

{{Isotopic uniform simplex polytopes}}

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

{{Octaexon family}}

• List of A7 polytopes

• 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

• [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