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Stericated 8-simplexes#Steriruncinated 8-simplex

{{Short description|Class of eight-dimensional polytopes}}

{{Lead too short|date=September 2024}}

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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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Stericated 8-simplex
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Bistericated 8-simplex
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Steri-truncated 8-simplex
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Bisteri-truncated 8-simplex
{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node|3|node}}

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Steri-cantellated 8-simplex
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Bisteri-cantellated 8-simplex
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Stericanti-truncated 8-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}

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

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Steri-runcinated 8-simplex
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Bisteri-runcinated 8-simplex
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Steriruncitruncated 8-simplex
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}

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

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

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

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

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Bisteriruncicanti-truncated 8-simplex
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colspan=4|Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

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Stericated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,4{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|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;"|Edges6300
style="background:#e7dcc3;"|Vertices630
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Coordinates =

The Cartesian coordinates of the vertices of the stericated 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 stericated 9-orthoplex.

= Images =

{{8-simplex Coxeter plane graphs|t04|120}}

Bistericated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt1,5{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_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;"|Edges12600
style="background:#e7dcc3;"|Vertices1260
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t15|120}}

Steritruncated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,4{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|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;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Images =

{{8-simplex Coxeter plane graphs|t014|120}}

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Bisteritruncated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt1,2,5{3,3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|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;"|Edges
style="background:#e7dcc3;"|Vertices
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Images =

{{8-simplex Coxeter plane graphs|t125|120}}

Stericantellated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t024|120}}

Bistericantellated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t135|120}}

Stericantitruncated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t0124|120}}

Bistericantitruncated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t1235|120}}

Steriruncinated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t034|120}}

Bisteriruncinated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t145|120}}

Steriruncitruncated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t0134|120}}

Bisteriruncitruncated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t1245|120}}

Steriruncicantellated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t0234|120}}

Bisteriruncicantellated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t1345|120}}

Steriruncicantitruncated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t01234|120}}

Bisteriruncicantitruncated 8-simplex

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

= Images =

{{8-simplex Coxeter plane graphs|t12345|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)}} x3o3o3o3x3o3o3o, o3x3o3o3o3x3o3o

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