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Pentellated 8-simplexes#Bipentellated 8-simplex

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

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

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Bipentitruncated 8-simplex
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colspan=3|Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex.

There are two unique pentellations of the 8-simplex. Including truncations, cantellations, runcinations, and sterications, there are 32 more pentellations. These polytopes are a part of a family 135 uniform 8-polytopes with A8 symmetry. A8, [37] has order 9 factorial symmetry, or 362880. The bipentalled form is symmetrically ringed, doubling the symmetry order to 725760, and is represented the double-bracketed group 37. The A8 Coxeter plane projection shows order [9] symmetry for the pentellated 8-simplex, while the bipentellated 8-simple is doubled to [18] symmetry.

Pentellated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,5{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node|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;"|Edges5040
style="background:#e7dcc3;"|Vertices504
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

Acronym: sotane (Jonathan Bowers)Klitzing, (x3o3o3o3o3x3o3o – sotane)

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t05|120}}

Bipentellated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt1,6{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_1|3|node}}
style="background:#e7dcc3;"|7-facest0,5{3,3,3,3,3,3}
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;"|Edges7560
style="background:#e7dcc3;"|Vertices756
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8×2, 37, order 725760
style="background:#e7dcc3;"|Propertiesconvex, facet-transitive

= Alternate names =

  • Small biterated bienneazetton
  • Bipentellated enneazetton (Acronym: sobteb) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sobteb.htm (o3x3o3o3o3o3x3o – sobteb)]}}

= Coordinates =

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

= Images =

{{8-simplex2 Coxeter plane graphs|t16|120}}

Related polytopes

Pentellated 8-simplex and dipentellated 8-simplex are two polytopes selected from 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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 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|polyzetta.htm|8D uniform polytopes (polyzetta) with acronyms}} x3o3o3o3o3x3o3o – sotane, o3x3o3o3o3o3x3o – sobteb {{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:8-polytopes