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Truncated 6-simplexes#Bitruncated 6-simplex

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|150px
6-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}

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Truncated 6-simplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node}}

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Bitruncated 6-simplex
{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node}}

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Tritruncated 6-simplex
{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node}}

colspan=3|Orthogonal projections in A7 Coxeter plane

In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex.

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Truncated 6-simplex

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!bgcolor=#e7dcc3 colspan=2|Truncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|ClassA6 polytope
bgcolor=#e7dcc3|Schläfli symbolt{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node}}
{{CDD|branch_11|3b|nodeb|3b|nodeb|3b|nodeb|3b|nodeb}}
bgcolor=#e7dcc3|5-faces14:
7 {3,3,3,3} 30px
7 t{3,3,3,3} 30px
bgcolor=#e7dcc3|4-faces63:
42 {3,3,3} 30px
21 t{3,3,3} 30px
bgcolor=#e7dcc3|Cells140:
105 {3,3} 30px
35 t{3,3} 30px
bgcolor=#e7dcc3|Faces175:
140 {3}
35 {6}
bgcolor=#e7dcc3|Edges126
bgcolor=#e7dcc3|Vertices42
bgcolor=#e7dcc3|Vertex figure60px
( )v{3,3,3}
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Dual?
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Truncated heptapeton (Acronym: til) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/til.htm (o3x3o3o3o3o - til)]}}

= Coordinates =

The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex.

= Images =

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

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Bitruncated 6-simplex

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!bgcolor=#e7dcc3 colspan=2|Bitruncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|ClassA6 polytope
bgcolor=#e7dcc3|Schläfli symbol2t{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node}}
{{CDD|branch_11|3ab|nodes|3b|nodeb|3b|nodeb}}
bgcolor=#e7dcc3|5-faces14
bgcolor=#e7dcc3|4-faces84
bgcolor=#e7dcc3|Cells245
bgcolor=#e7dcc3|Faces385
bgcolor=#e7dcc3|Edges315
bgcolor=#e7dcc3|Vertices105
bgcolor=#e7dcc3|Vertex figure60px
{ }v{3,3}
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/batal.htm (o3x3x3o3o3o - batal)]}}

= Coordinates =

The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 7-orthoplex.

= Images =

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

Tritruncated 6-simplex

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!bgcolor=#e7dcc3 colspan=2|Tritruncated 6-simplex

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|ClassA6 polytope
bgcolor=#e7dcc3|Schläfli symbol3t{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDDnode|3|node3|node_1|3|node_1|3|node|3|node}}
or {{CDD|branch_11|3ab|nodes|3ab|nodes}}
bgcolor=#e7dcc3|5-faces14 2t{3,3,3,3}
bgcolor=#e7dcc3|4-faces84
bgcolor=#e7dcc3|Cells280
bgcolor=#e7dcc3|Faces490
bgcolor=#e7dcc3|Edges420
bgcolor=#e7dcc3|Vertices140
bgcolor=#e7dcc3|Vertex figure60px
{3}v{3}
bgcolor=#e7dcc3|Coxeter groupA6, 35, order 10080
bgcolor=#e7dcc3|Propertiesconvex, isotopic

The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets.

The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration: {{CDD|branch|3ab|nodes|3ab|nodes_10l}} and {{CDD|branch|3ab|nodes|3ab|nodes_01l}}.

= Alternate names =

  • Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/fe.htm (o3o3x3x3o3o - fe)]}}

= Coordinates =

The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,2). This construction is based on facets of the bitruncated 7-orthoplex. Alternately it can be centered on the origin as permutations of (-1,-1,-1,0,1,1,1).

= Images =

{{6-simplex2 Coxeter plane graphs|t23|150}}

= Related polytopes =

{{Isotopic uniform simplex polytopes}}

Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

{{Heptapeton 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|polypeta.htm|6D uniform polytopes (polypeta) with acronyms}} o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe {{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:6-polytopes