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Truncated 6-orthoplexes#Truncated 6-orthoplex

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6-orthoplex
{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}

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Truncated 6-orthoplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node|4|node}}

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Bitruncated 6-orthoplex
{{CDD|node|3|node_1|3|node_1|3|node|3|node|4|node}}

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Tritruncated 6-cube
{{CDD|node|3|node|3|node_1|3|node_1|3|node|4|node}}

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6-cube
{{CDD|node|3|node|3|node|3|node|3|node|4|node_1}}

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Truncated 6-cube
{{CDD|node|3|node|3|node|3|node|3|node_1|4|node_1}}

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Bitruncated 6-cube
{{CDD|node|3|node|3|node|3|node_1|3|node_1|4|node}}

colspan=4|Orthogonal projections in B6 Coxeter plane

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

There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.

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Truncated 6-orthoplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt{3,3,3,3,4}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node|4|node}}

{{CDD|node_1|3|node_1|3|node|3|node|split1|nodes}}

bgcolor=#e7dcc3|5-faces76
bgcolor=#e7dcc3|4-faces576
bgcolor=#e7dcc3|Cells1200
bgcolor=#e7dcc3|Faces1120
bgcolor=#e7dcc3|Edges540
bgcolor=#e7dcc3|Vertices120
bgcolor=#e7dcc3|Vertex figure80px
( )v{3,4}
bgcolor=#e7dcc3|Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Truncated hexacross
  • Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/tag.htm (x3x3o3o3o4o - tag)]}}

= Construction =

There are two Coxeter groups associated with the truncated hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

= Coordinates =

Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of

: (±2,±1,0,0,0,0)

= Images =

{{6-cube Coxeter plane graphs|t45|150}}

Bitruncated 6-orthoplex

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbol2t{3,3,3,3,4}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node_1|3|node|3|node|4|node}}

{{CDD|node|3|node_1|3|node_1|3|node|split1|nodes}}

bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges
bgcolor=#e7dcc3|Vertices
bgcolor=#e7dcc3|Vertex figure80px
{ }v{3,4}
bgcolor=#e7dcc3|Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Bitruncated hexacross
  • Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/botag.htm (o3x3x3o3o4o - botag)]}}

= Images =

{{6-cube Coxeter plane graphs|t34|150}}

Related polytopes

These polytopes are a part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

{{Hexeract 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}} x3x3o3o3o4o - tag, o3x3x3o3o4o - botag {{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