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Truncated 7-simplexes#Bitruncated 7-simplex

{{Short description|Uniform 7-polytope}}

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|180px
7-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}

|180px
Truncated 7-simplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node}}

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Bitruncated 7-simplex
{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}

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Tritruncated 7-simplex
{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}

colspan=3|Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.

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Truncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Truncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt{3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node}}
style="background:#e7dcc3;"|6-faces16
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells350
style="background:#e7dcc3;"|Faces336
style="background:#e7dcc3;"|Edges196
style="background:#e7dcc3;"|Vertices56
style="background:#e7dcc3;"|Vertex figure( )v{3,3,3,3}
style="background:#e7dcc3;"|Coxeter groupsA7, [3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex, Vertex-transitive

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

= Alternate names =

  • Truncated octaexon (Acronym: toc) (Jonathan Bowers)Klitizing, (x3x3o3o3o3o3o - toc)

= Coordinates =

The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex.

= Images =

{{7-simplex Coxeter plane graphs|t01|150}}

Bitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Bitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbol2t{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|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;"|Edges588
style="background:#e7dcc3;"|Vertices168
style="background:#e7dcc3;"|Vertex figure{ }v{3,3,3}
style="background:#e7dcc3;"|Coxeter groupsA7, [3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex, Vertex-transitive

= Alternate names =

  • Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)Klitizing, (o3x3x3o3o3o3o - roc)

= Coordinates =

The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 8-orthoplex.

= Images =

{{7-simplex Coxeter plane graphs|t12|150}}

Tritruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Tritruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbol3t{3,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|3|node|3|node_1|3|node_1|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;"|Edges980
style="background:#e7dcc3;"|Vertices280
style="background:#e7dcc3;"|Vertex figure{3}v{3,3}
style="background:#e7dcc3;"|Coxeter groupsA7, [3,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex, Vertex-transitive

= Alternate names =

  • Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)Klitizing, (o3o3x3x3o3o3o - tattoc)

= Coordinates =

The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 8-orthoplex.

= Images =

{{7-simplex Coxeter plane graphs|t23|150}}

Related polytopes

These three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry.

{{Octaexon family}}

See also

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|polyexa.htm|7D|uniform polytopes (polyexa)}} x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc

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:7-polytopes