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Rectified 8-simplexes#Birectified 8-simplex

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8-simplex

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Rectified 8-simplex

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Birectified 8-simplex

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Trirectified 8-simplex

colspan=4|Orthogonal projections in A8 Coxeter plane

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

There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.

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Rectified 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
bgcolor=#e7dcc3|Coxeter symbol061
style="background:#e7dcc3;"|Schläfli symbolt1{37}
r{37} = {36,1}
or \left\{\begin{array}{l}3, 3, 3, 3, 3,3\\3\end{array}\right\}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
or {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
style="background:#e7dcc3;"|7-faces18
style="background:#e7dcc3;"|6-faces108
style="background:#e7dcc3;"|5-faces336
style="background:#e7dcc3;"|4-faces630
style="background:#e7dcc3;"|Cells756
style="background:#e7dcc3;"|Faces588
style="background:#e7dcc3;"|Edges252
style="background:#e7dcc3;"|Vertices36
style="background:#e7dcc3;"|Vertex figure7-simplex prism, {}×{3,3,3,3,3}
style="background:#e7dcc3;"|Petrie polygonenneagon
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|8}}. It is also called 06,1 for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t1|120}}

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Birectified 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
bgcolor=#e7dcc3|Coxeter symbol052
style="background:#e7dcc3;"|Schläfli symbolt2{37}
2r{37} = {35,2} or
\left\{\begin{array}{l}3, 3, 3, 3, 3\\3, 3\end{array}\right\}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
or {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea|3a|nodea}}
style="background:#e7dcc3;"|7-faces18
style="background:#e7dcc3;"|6-faces144
style="background:#e7dcc3;"|5-faces588
style="background:#e7dcc3;"|4-faces1386
style="background:#e7dcc3;"|Cells2016
style="background:#e7dcc3;"|Faces1764
style="background:#e7dcc3;"|Edges756
style="background:#e7dcc3;"|Vertices84
style="background:#e7dcc3;"|Vertex figure{3}×{3,3,3,3}
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|2|8}}. It is also called 05,2 for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea|3a|nodea}}.

The birectified 8-simplex is the vertex figure of the 152 honeycomb.

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t2|120}}

Trirectified 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
bgcolor=#e7dcc3|Coxeter symbol043
style="background:#e7dcc3;"|Schläfli symbolt3{37}
3r{37} = {34,3} or
\left\{\begin{array}{l}3, 3, 3, 3\\3, 3,3 \end{array}\right\}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|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|3a|nodea}}
style="background:#e7dcc3;"|7-faces9 + 9
style="background:#e7dcc3;"|6-faces36 + 72 + 36
style="background:#e7dcc3;"|5-faces84 + 252 + 252 + 84
style="background:#e7dcc3;"|4-faces126 + 504 + 756 + 504
style="background:#e7dcc3;"|Cells630 + 1260 + 1260
style="background:#e7dcc3;"|Faces1260 + 1680
style="background:#e7dcc3;"|Edges1260
style="background:#e7dcc3;"|Vertices126
style="background:#e7dcc3;"|Vertex figure{3,3}×{3,3,3}
style="background:#e7dcc3;"|Petrie polygonenneagon
style="background:#e7dcc3;"|Coxeter groupA7, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|3|8}}. It is also called 04,3 for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3a|nodea}}.

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t3|120}}

Related polytopes

This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb.

It is also 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)}} o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene

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