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Rectified 6-cubes#Rectified 6-cube

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

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

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

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

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

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

colspan=4|Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.

There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-cube are located in the square face centers of the 6-cube.

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Rectified 6-cube

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!bgcolor=#e7dcc3 colspan=2|Rectified 6-cube

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt1{4,34} or r{4,34}
\left\{\begin{array}{l}4\\3, 3, 3, 3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node|3|node|3|node|3|node}} = {{CDD|node_1|split1-43|nodes|3b|nodeb|3b|nodeb|3b|nodeb}}
{{CDD|nodes_11|split2|node|3|node|3|node|3|node}}
bgcolor=#e7dcc3|5-faces76
bgcolor=#e7dcc3|4-faces444
bgcolor=#e7dcc3|Cells1120
bgcolor=#e7dcc3|Faces1520
bgcolor=#e7dcc3|Edges960
bgcolor=#e7dcc3|Vertices192
bgcolor=#e7dcc3|Vertex figure5-cell prism
bgcolor=#e7dcc3|Petrie polygonDodecagon
bgcolor=#e7dcc3|Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Rectified hexeract (acronym: rax) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/rax.htm (o3o3o3o3x4o - rax)]}}

= Construction =

The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

= Coordinates =

The Cartesian coordinates of the vertices of the rectified 6-cube with edge length {{radic|2}} are all permutations of:

:(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm1)

= Images =

{{6-cube Coxeter plane graphs|t1|150}}

Birectified 6-cube

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!bgcolor=#e7dcc3 colspan=2|Birectified 6-cube

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Coxeter symbol0311
bgcolor=#e7dcc3|Schläfli symbolt2{4,34} or 2r{4,34}
\left\{\begin{array}{l}3, 4\\3, 3, 3\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node|3|node_1|3|node|3|node|3|node}} = {{CDD|node_1|split1|nodes|4a3b|nodes|3b|nodeb}}
{{CDD|nodes|split2|node_1|3|node|3|node|3|node}} = {{CDD|nodes|3ab|nodes_01lr|split5c|nodes}}
bgcolor=#e7dcc3|5-faces76
bgcolor=#e7dcc3|4-faces636
bgcolor=#e7dcc3|Cells2080
bgcolor=#e7dcc3|Faces3200
bgcolor=#e7dcc3|Edges1920
bgcolor=#e7dcc3|Vertices240
bgcolor=#e7dcc3|Vertex figure{4}x{3,3} duoprism
bgcolor=#e7dcc3|Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Birectified hexeract (acronym: brox) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/brox.htm (o3o3o3x3o4o - brox)]}}
  • Rectified 6-demicube

= Construction =

The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

= Coordinates =

The Cartesian coordinates of the vertices of the rectified 6-cube with edge length {{radic|2}} are all permutations of:

:(0,\ 0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)

= Images =

{{6-cube Coxeter plane graphs|t2|150}}

Related polytopes

These polytopes are 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, [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}} o3o3o3o3x4o - rax, o3o3o3x3o4o - brox {{sfn whitelist| CITEREFKlitzing}}

External links

  • {{MathWorld|title=Hypercube|urlname=Hypercube}}
  • [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