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Rectified 9-simplexes#Birectified 9-simplex

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

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Rectified 9-simplex
{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}

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Birectified 9-simplex
{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}

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Trirectified 9-simplex
{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}

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Quadrirectified 9-simplex
{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}

colspan=4|Orthogonal projections in A9 Coxeter plane

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.

These polytopes are part of a family of 271 uniform 9-polytopes with A9 symmetry.

There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the tetrahedral cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex.

Rectified 9-simplex

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! style="background:#e7dcc3;" colspan="2"|Rectified 9-simplex

style="background:#e7dcc3;"|Typeuniform 9-polytope
style="background:#e7dcc3;"|Schläfli symbolt1{3,3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
style="background:#e7dcc3;"|8-faces20
style="background:#e7dcc3;"|7-faces135
style="background:#e7dcc3;"|6-faces480
style="background:#e7dcc3;"|5-faces1050
style="background:#e7dcc3;"|4-faces1512
style="background:#e7dcc3;"|Cells1470
style="background:#e7dcc3;"|Faces960
style="background:#e7dcc3;"|Edges360
style="background:#e7dcc3;"|Vertices45
style="background:#e7dcc3;"|Vertex figure8-simplex prism
style="background:#e7dcc3;"|Petrie polygondecagon
style="background:#e7dcc3;"|Coxeter groupsA9, [3,3,3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

The rectified 9-simplex is the vertex figure of the 10-demicube.

= Alternate names=

  • Rectified decayotton (reday) (Jonathan Bowers)Klitzing, (o3x3o3o3o3o3o3o3o - reday)

= Coordinates =

The Cartesian coordinates of the vertices of the rectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 10-orthoplex.

= Images =

{{A9 Coxeter plane graphs|t1|100}}

Birectified 9-simplex

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! style="background:#e7dcc3;" colspan="2"|Birectified 9-simplex

style="background:#e7dcc3;"|Typeuniform 9-polytope
style="background:#e7dcc3;"|Schläfli symbolt2{3,3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
style="background:#e7dcc3;"|8-faces
style="background:#e7dcc3;"|7-faces
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;"|Edges1260
style="background:#e7dcc3;"|Vertices120
style="background:#e7dcc3;"|Vertex figure{3}×{3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter groupsA9, [3,3,3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

This polytope is the vertex figure for the 162 honeycomb. Its 120 vertices represent the kissing number of the related hyperbolic 9-dimensional sphere packing.

= Alternate names=

  • Birectified decayotton (breday) (Jonathan Bowers)Klitzing, (o3o3x3o3o3o3o3o3o - breday)

= Coordinates =

The Cartesian coordinates of the vertices of the birectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 10-orthoplex.

= Images =

{{A9 Coxeter plane graphs|t2|100}}

Trirectified 9-simplex

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! style="background:#e7dcc3;" colspan="2"|Trirectified 9-simplex

style="background:#e7dcc3;"|Typeuniform 9-polytope
style="background:#e7dcc3;"|Schläfli symbolt3{3,3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
style="background:#e7dcc3;"|8-faces
style="background:#e7dcc3;"|7-faces
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;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure{3,3}×{3,3,3,3}
style="background:#e7dcc3;"|Coxeter groupsA9, [3,3,3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Trirectified decayotton (treday) (Jonathan Bowers)Klitzing, (o3o3o3x3o3o3o3o3o - treday)

= Coordinates =

The Cartesian coordinates of the vertices of the trirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 10-orthoplex.

= Images =

{{A9 Coxeter plane graphs|t3|100}}

Quadrirectified 9-simplex

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! style="background:#e7dcc3;" colspan="2"|Quadrirectified 9-simplex

style="background:#e7dcc3;"|Typeuniform 9-polytope
style="background:#e7dcc3;"|Schläfli symbolt4{3,3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
or {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes}}
style="background:#e7dcc3;"|8-faces
style="background:#e7dcc3;"|7-faces
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;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure{3,3,3}×{3,3,3}
style="background:#e7dcc3;"|Coxeter groupsA9×2, 38
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Quadrirectified decayotton
  • Icosayotton (icoy) (Jonathan Bowers)Klitzing, (o3o3o3o3x3o3o3o3o - icoy)

= Coordinates =

The Cartesian coordinates of the vertices of the quadrirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 10-orthoplex.

= Images =

{{A9 Coxeter plane graphs|t4|100}}

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. (1966)
  • {{KlitzingPolytopes|polyyotta.htm|9D|uniform polytopes (polyyotta)}} o3x3o3o3o3o3o3o3o - reday, o3o3x3o3o3o3o3o3o - breday, o3o3o3x3o3o3o3o3o - treday, o3o3o3o3x3o3o3o3o - icoy

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:9-polytopes