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Cantellated 8-simplexes#Bicantitruncated 8-simplex

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

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

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Tricantellated
8-simplex
{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}

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

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

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

colspan=4|Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.

There are six unique cantellations for the 8-simplex, including permutations of truncation.

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Cantellated 8-simplex

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! style="background:#e7dcc3;" colspan="2"|Cantellated 8-simplex

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolrr{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagram{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
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;"|Edges1764
style="background:#e7dcc3;"|Vertices252
style="background:#e7dcc3;"|Vertex figure6-simplex prism
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)Klitizing, (x3o3x3o3o3o3o3o - srene)

= Coordinates =

The Cartesian coordinates of the vertices of the cantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 9-orthoplex.

= Images =

{{8-simplex Coxeter plane graphs|t02|120}}

Bicantellated 8-simplex

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! style="background:#e7dcc3;" colspan="2"|Bicantellated 8-simplex

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolr2r{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagram{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node}}
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;"|Edges5292
style="background:#e7dcc3;"|Vertices756
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)Klitizing, (o3x3o3x3o3o3o3o - sabrene)

= Coordinates =

The Cartesian coordinates of the vertices of the bicantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 9-orthoplex.

= Images =

{{8-simplex Coxeter plane graphs|t13|120}}

Tricantellated 8-simplex

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! style="background:#e7dcc3;" colspan="2"|tricantellated 8-simplex

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolr3r{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagram{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}
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;"|Edges8820
style="background:#e7dcc3;"|Vertices1260
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)Klitizing, (o3o3x3o3x3o3o3o - satrene)

= Coordinates =

The Cartesian coordinates of the vertices of the tricantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the tricantellated 9-orthoplex.

= Images =

{{8-simplex Coxeter plane graphs|t13|120}}

Cantitruncated 8-simplex

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! style="background:#e7dcc3;" colspan="2"|Cantitruncated 8-simplex

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symboltr{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagram{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node}}
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
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)Klitizing, (x3x3x3o3o3o3o3o - grene)

= Coordinates =

The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

= Images =

{{8-simplex Coxeter plane graphs|t012|120}}

Bicantitruncated 8-simplex

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! style="background:#e7dcc3;" colspan="2"|Bicantitruncated 8-simplex

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt2r{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagram{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}
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
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

Alternate names

  • Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)Klitizing, (o3x3x3x3o3o3o3o - gabrene)

= Coordinates =

The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

= Images =

{{8-simplex Coxeter plane graphs|t123|120}}

Tricantitruncated 8-simplex

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! style="background:#e7dcc3;" colspan="2"|Tricantitruncated 8-simplex

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt3r{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagram{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
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
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

  • Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)Klitizing, (o3o3x3x3x3o3o3o - gatrene)

= Coordinates =

The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

= Images =

{{8-simplex Coxeter plane graphs|t234|120}}

Related polytopes

This polytope is one of 135 uniform 8-polytopes with A8 symmetry.

{{Enneazetton 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, {{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|polyzetta.htm|8D|uniform polytopes (polyzetta)}} x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene

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:8-polytopes