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Runcinated 8-simplexes#Runcicantellated 8-simplex

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8-simplex
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Runcinated 8-simplex
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Biruncinated 8-simplex
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Triruncinated 8-simplex
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Runcitruncated 8-simplex
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Biruncitruncated 8-simplex
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Triruncitruncated 8-simplex
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Runcicantellated 8-simplex
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Biruncicantellated 8-simplex
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Runcicantitruncated 8-simplex
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Biruncicantitruncated 8-simplex
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Triruncicantitruncated 8-simplex
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colspan=4|Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicanti{{wbr}}truncated 8-simplex have a doubled symmetry, showing [18] order reflectional symmetry in the A8 Coxeter plane.

Runcinated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,3{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
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;"|Edges4536
style="background:#e7dcc3;"|Vertices504
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

=Alternate names =

  • Runcinated enneazetton
  • Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)Klitzing (x3o3o3x3o3o3o3o - spene)

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t03|120}}

Biruncinated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt1,4{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagram{{CDD|node|3|node_1|3|node|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;"|Edges11340
style="background:#e7dcc3;"|Vertices1260
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

=Alternate names =

  • Biruncinated enneazetton
  • Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)Klitzing (o3x3o3o3x3o3o3o - sabpene)

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t14|120}}

Triruncinated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt2,5{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_1|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;"|Edges15120
style="background:#e7dcc3;"|Vertices1680
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8×2, 37, order 725760
style="background:#e7dcc3;"|Propertiesconvex

=Alternate names =

  • Triruncinated enneazetton
  • Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)Klitzing (o3o3x3o3o3x3o3o - satpeb)

= Coordinates =

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

= Images =

{{8-simplex2 Coxeter plane graphs|t25|120}}

Runcitruncated 8-simplex

{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node}}

= Images =

{{8-simplex2 Coxeter plane graphs|t013|120}}

Biruncitruncated 8-simplex

{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}

= Images =

{{8-simplex2 Coxeter plane graphs|t124|120}}

Triruncitruncated 8-simplex

{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}

= Images =

{{8-simplex2 Coxeter plane graphs|t235|120}}

Runcicantellated 8-simplex

{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}

= Images =

{{8-simplex2 Coxeter plane graphs|t023|120}}

Biruncicantellated 8-simplex

{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}

= Images =

{{8-simplex2 Coxeter plane graphs|t134|120}}

Runcicantitruncated 8-simplex

{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}

= Images =

{{8-simplex2 Coxeter plane graphs|t0123|120}}

Biruncicantitruncated 8-simplex

{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}

= Images =

{{8-simplex2 Coxeter plane graphs|t1234|120}}

Triruncicantitruncated 8-simplex

{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}

= Images =

{{8-simplex2 Coxeter plane graphs|t2345|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)}} x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb

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