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Runcinated 6-simplexes#Biruncitruncated 6-simplex

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

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

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

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

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

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

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

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

colspan=3|Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.

There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.

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Runcinated 6-simplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,3{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|5-faces70
bgcolor=#e7dcc3|4-faces455
bgcolor=#e7dcc3|Cells1330
bgcolor=#e7dcc3|Faces1610
bgcolor=#e7dcc3|Edges840
bgcolor=#e7dcc3|Vertices140
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Small prismated heptapeton (Acronym: spil) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/spil.htm (x3o3o3x3o3o - spil)]}}

= Coordinates =

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

= Images =

{{6-simplex Coxeter plane graphs|t03|150}}

Biruncinated 6-simplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt1,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces84
bgcolor=#e7dcc3|4-faces714
bgcolor=#e7dcc3|Cells2100
bgcolor=#e7dcc3|Faces2520
bgcolor=#e7dcc3|Edges1260
bgcolor=#e7dcc3|Vertices210
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, 35, order 10080
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sibpof.htm (o3x3o3o3x3o - sibpof)]}}

= Coordinates =

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

= Images =

{{6-simplex2 Coxeter plane graphs|t14|150}}

Runcitruncated 6-simplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,3{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|5-faces70
bgcolor=#e7dcc3|4-faces560
bgcolor=#e7dcc3|Cells1820
bgcolor=#e7dcc3|Faces2800
bgcolor=#e7dcc3|Edges1890
bgcolor=#e7dcc3|Vertices420
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/patal.htm (x3x3o3x3o3o - patal)]}}

= Coordinates =

The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t013|150}}

Biruncitruncated 6-simplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt1,2,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces84
bgcolor=#e7dcc3|4-faces714
bgcolor=#e7dcc3|Cells2310
bgcolor=#e7dcc3|Faces3570
bgcolor=#e7dcc3|Edges2520
bgcolor=#e7dcc3|Vertices630
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/bapril.htm (o3x3x3o3x3o - bapril)]}}

= Coordinates =

The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t124|150}}

Runcicantellated 6-simplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,2,3{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|5-faces70
bgcolor=#e7dcc3|4-faces455
bgcolor=#e7dcc3|Cells1295
bgcolor=#e7dcc3|Faces1960
bgcolor=#e7dcc3|Edges1470
bgcolor=#e7dcc3|Vertices420
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/pril.htm (x3o3x3x3o3o - pril)]}}

= Coordinates =

The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t023|150}}

Runcicantitruncated 6-simplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,2,3{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|5-faces70
bgcolor=#e7dcc3|4-faces560
bgcolor=#e7dcc3|Cells1820
bgcolor=#e7dcc3|Faces3010
bgcolor=#e7dcc3|Edges2520
bgcolor=#e7dcc3|Vertices840
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, [35], order 5040
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Runcicantitruncated heptapeton
  • Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/gapil.htm (x3x3x3x3o3o - gapil)]}}

= Coordinates =

The vertices of the runcicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 7-orthoplex.

= Images =

{{6-simplex Coxeter plane graphs|t0123|150}}

Biruncicantitruncated 6-simplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt1,2,3,4{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces84
bgcolor=#e7dcc3|4-faces714
bgcolor=#e7dcc3|Cells2520
bgcolor=#e7dcc3|Faces4410
bgcolor=#e7dcc3|Edges3780
bgcolor=#e7dcc3|Vertices1260
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupA6, 35, order 10080
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Biruncicantitruncated heptapeton
  • Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/gibpof.htm (o3x3x3x3x3o - gibpof)]}}

= Coordinates =

The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 7-orthoplex.

= Images =

{{6-simplex2 Coxeter plane graphs|t1234|150}}

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, [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html 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}} x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof {{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