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Cantellated 7-simplexes

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

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

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

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Tricantellated 7-simplex
{{CDD|node|3|node|3|node_1|3|node|3|node_1|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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Cantitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}

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

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

colspan=4|Orthogonal projections in A7 Coxeter plane

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In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.

There are unique 6 degrees of cantellation for the 7-simplex, including truncations.

Cantellated 7-simplex

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

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbolrr{3,3,3,3,3,3}
or r\left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node}}
or {{CDD|node|split1|nodes_11		3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
bgcolor=#e7dcc3|6-faces
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges1008
bgcolor=#e7dcc3|Vertices168
bgcolor=#e7dcc3|Vertex figure5-simplex prism
bgcolor=#e7dcc3|Coxeter groupsA7, [3,3,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Small rhombated octaexon (acronym: saro) (Jonathan Bowers)Klitizing, (x3o3x3o3o3o3o - saro)

= Coordinates =

The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.

= Images =

{{7-simplex Coxeter plane graphs|t02|150}}

Bicantellated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Bicantellated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolr2r{3,3,3,3,3,3}
or r\left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}
or {{CDD|node|split1|nodes_11|3ab|nodes|3a|nodea|3a|nodea}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges2520
style="background:#e7dcc3;"|Vertices420
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsA7, [3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)Klitizing, (o3x3o3x3o3o3o - sabro)

= Coordinates =

The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.

= Images =

{{7-simplex Coxeter plane graphs|t13|150}}

Tricantellated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Tricantellated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolr3r{3,3,3,3,3,3}
or r\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}
or {{CDD|node|split1|nodes_11|3ab|nodes|3ab|nodes}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges3360
style="background:#e7dcc3;"|Vertices560
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsA7, [3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)Klitizing, (o3o3x3o3x3o3o - stiroh)

= Coordinates =

The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.

= Images =

{{7-simplex Coxeter plane graphs|t24|150}}

Cantitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Cantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symboltr{3,3,3,3,3,3}
or t\left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}
{{CDD|node_1|split1|nodes_11		3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges1176
style="background:#e7dcc3;"|Vertices336
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsA7, [3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Great rhombated octaexon (acronym: garo) (Jonathan Bowers)Klitizing, (x3x3x3o3o3o3o - garo)

= Coordinates =

The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.

= Images =

{{7-simplex Coxeter plane graphs|t012|150}}

Bicantitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Bicantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt2r{3,3,3,3,3,3}
or t\left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
or {{CDD|node_1|split1|nodes_11|3ab|nodes|3a|nodea|3a|nodea}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges2940
style="background:#e7dcc3;"|Vertices840
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsA7, [3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)Klitizing, (o3x3x3x3o3o3o - gabro)

= Coordinates =

The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.

= Images =

{{7-simplex Coxeter plane graphs|t123|150}}

Tricantitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Tricantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt3r{3,3,3,3,3,3}
or t\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}
or {{CDD|node_1|split1|nodes_11|3ab|nodes|3ab|nodes}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges3920
style="background:#e7dcc3;"|Vertices1120
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsA7, [3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)Klitizing, (o3o3x3x3x3o3o - gatroh)

= Coordinates =

The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.

= Images =

{{7-simplex2 Coxeter plane graphs|t234|150}}

Related polytopes

This polytope is one of 71 uniform 7-polytopes with A7 symmetry.

{{Octaexon family}}

See also

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 Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • (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)}} x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh

External links

  • [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:7-polytopes