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Truncated 8-simplexes#Tritruncated 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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Truncated 8-simplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}

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

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

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

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

colspan=3|Orthogonal projections in A8 Coxeter plane

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

There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex.

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Truncated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbolt{37}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|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;"|Edges288
style="background:#e7dcc3;"|Vertices72
style="background:#e7dcc3;"|Vertex figure( )v{3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Truncated enneazetton (Acronym: tene) (Jonathan Bowers)Klitizing, (x3x3o3o3o3o3o3o - tene)

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t01|120}}

Bitruncated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbol2t{37}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|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;"|Edges1008
style="background:#e7dcc3;"|Vertices252
style="background:#e7dcc3;"|Vertex figure{ }v{3,3,3,3}
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)Klitizing, (o3x3x3o3o3o3o3o - batene)

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t12|120}}

Tritruncated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbol3t{37}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|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;"|Edges2016
style="background:#e7dcc3;"|Vertices504
style="background:#e7dcc3;"|Vertex figure{3}v{3,3,3}
style="background:#e7dcc3;"|Coxeter groupA8, [37], order 362880
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)Klitizing, (o3o3x3x3o3o3o3o - tatene)

= Coordinates =

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

= Images =

{{8-simplex Coxeter plane graphs|t23|120}}

Quadritruncated 8-simplex

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

style="background:#e7dcc3;"|Typeuniform 8-polytope
style="background:#e7dcc3;"|Schläfli symbol4t{37}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}
or {{CDD|branch_11|3ab|nodes|3ab|nodes|3ab|nodes}}
style="background:#e7dcc3;"|6-faces18 3t{3,3,3,3,3,3}
style="background:#e7dcc3;"|7-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges2520
style="background:#e7dcc3;"|Vertices630
style="background:#e7dcc3;"|Vertex figure80px
{3,3}v{3,3}
style="background:#e7dcc3;"|Coxeter groupA8, 37, order 725760
style="background:#e7dcc3;"|Propertiesconvex, isotopic

The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets.

= Alternate names=

  • Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)Klitizing, (o3o3o3x3x3o3o3o - be)

= Coordinates =

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

= Images =

{{8-simplex2 Coxeter plane graphs|t34|120}}

= Related polytopes=

{{Isotopic uniform simplex polytopes}}

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)}} x3x3o3o3o3o3o3o - tene, o3x3x3o3o3o3o3o - batene, o3o3x3x3o3o3o3o - tatene, o3o3o3x3x3o3o3o - be

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