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Cantellated 6-simplexes

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

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Cantellated 6-simplex
{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node}}

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Bicantellated 6-simplex
{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node}}

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Birectified 6-simplex
{{CDD|node|3|node|3|node_1|3|node|3|node|3|node}}

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Cantitruncated 6-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}

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Bicantitruncated 6-simplex
{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}

colspan=3|Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.

There are unique 4 degrees of cantellation for the 6-simplex, including truncations.

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Cantellated 6-simplex

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!bgcolor=#e7dcc3 colspan=2|Cantellated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolrr{3,3,3,3,3}
or r\left\{\begin{array}{l}3, 3, 3, 3\\3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node}}
{{CDD|node|split1|nodes_11|3a|nodea|3a|nodea|3a|nodea}}
bgcolor=#e7dcc3|5-faces35
bgcolor=#e7dcc3|4-faces210
bgcolor=#e7dcc3|Cells560
bgcolor=#e7dcc3|Faces805
bgcolor=#e7dcc3|Edges525
bgcolor=#e7dcc3|Vertices105
bgcolor=#e7dcc3|Vertex figure5-cell prism
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sril.htm (x3o3x3o3o3o - sril)]}}

= Coordinates =

The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.

= Images =

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

Bicantellated 6-simplex

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!bgcolor=#e7dcc3 colspan=2|Bicantellated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbol2rr{3,3,3,3,3}
or r\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node}}
{{CDD|node|split1|nodes_11|3ab|nodes|3a|nodea}}
bgcolor=#e7dcc3|5-faces49
bgcolor=#e7dcc3|4-faces329
bgcolor=#e7dcc3|Cells980
bgcolor=#e7dcc3|Faces1540
bgcolor=#e7dcc3|Edges1050
bgcolor=#e7dcc3|Vertices210
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sabril.htm (o3x3o3x3o3o - sabril)]}}

= Coordinates =

The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.

= Images =

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

Cantitruncated 6-simplex

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!bgcolor=#e7dcc3 colspan=2|cantitruncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symboltr{3,3,3,3,3}
or t\left\{\begin{array}{l}3, 3, 3, 3\\3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
{{CDD|node_1|split1|nodes_11|3a|nodea|3a|nodea|3a|nodea}}
bgcolor=#e7dcc3|5-faces35
bgcolor=#e7dcc3|4-faces210
bgcolor=#e7dcc3|Cells560
bgcolor=#e7dcc3|Faces805
bgcolor=#e7dcc3|Edges630
bgcolor=#e7dcc3|Vertices210
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/gril.htm (x3x3x3o3o3o - gril)]}}

= Coordinates =

The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.

= Images =

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

Bicantitruncated 6-simplex

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!bgcolor=#e7dcc3 colspan=2|bicantitruncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbol2tr{3,3,3,3,3}
or t\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}
{{CDD|node_1|split1|nodes_11|3ab|nodes|3a|nodea}}
bgcolor=#e7dcc3|5-faces49
bgcolor=#e7dcc3|4-faces329
bgcolor=#e7dcc3|Cells980
bgcolor=#e7dcc3|Faces1540
bgcolor=#e7dcc3|Edges1260
bgcolor=#e7dcc3|Vertices420
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/gabril.htm (o3x3x3x3o3o - gabril)]}}

= Coordinates =

The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.

= Images =

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

Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 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 Ivic Weiss, Wiley-Interscience Publication, 1995, [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html 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}} x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril {{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