Login
Login

Cantellated 6-cubes#Cantellated 6-cube

class=wikitable align=right width=360 style="margin-left:1em;"
align=center

|colspan=4|120px
6-cube
{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}

|colspan=4|120px
Cantellated 6-cube
{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node}}

|colspan=4|120px
Bicantellated 6-cube
{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node}}

align=center

|colspan=4|120px
6-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node|3|node_1}}

|colspan=4|120px
Cantellated 6-orthoplex
{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1}}

|colspan=4|120px
Bicantellated 6-orthoplex
{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node}}

align=center

|colspan=3|90px
Cantitruncated 6-cube
{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node}}

|colspan=3|90px
Bicantitruncated 6-cube
{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node}}

|colspan=3|90px
Bicantitruncated 6-orthoplex
{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node}}

|colspan=3|90px
Cantitruncated 6-orthoplex
{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1}}

colspan=12|Orthogonal projections in B6 Coxeter plane

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

There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual 6-orthoplex.

{{TOC left}}

{{Clear}}

Cantellated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolrr{4,3,3,3,3}
or r\left\{\begin{array}{l}3, 3, 3, 3\\4\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node}}
{{CDD|node|split1-43|nodes_11|3b|nodeb|3b|nodeb|3b|nodeb}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges4800
bgcolor=#e7dcc3|Vertices960
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsB6, [3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Cantellated hexeract
  • Small rhombated hexeract (acronym: srox) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/srox.htm (o3o3o3x3o4x - srox)]}}

= Images =

{{6-cube Coxeter plane graphs|t02|150}}

Bicantellated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbol2rr{4,3,3,3,3}
or r\left\{\begin{array}{l}3, 3, 3\\3, 4\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node}}
{{CDD|node|split1|nodes_11|4a3b|nodes|3b|nodeb}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges
bgcolor=#e7dcc3|Vertices
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsB6, [3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Bicantellated hexeract
  • Small birhombated hexeract (acronym: saborx) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/saborx.htm (o3o3x3o3x4o - saborx)]}}

= Images =

{{6-cube Coxeter plane graphs|t13|150}}

Cantitruncated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symboltr{4,3,3,3,3}
or t\left\{\begin{array}{l}3, 3, 3, 3\\4\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node}}
{{CDD|node_1|split1-43|nodes_11|3b|nodeb|3b|nodeb|3b|nodeb}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges
bgcolor=#e7dcc3|Vertices
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsB6, [3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Cantitruncated hexeract
  • Great rhombihexeract (acronym: grox) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/grox.htm (o3o3o3x3x4x - grox)]}}

= Images =

{{6-cube Coxeter plane graphs|t012|150}}

It is fourth in a series of cantitruncated hypercubes:

{{Cantitruncated hypercube polytopes}}

Bicantitruncated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbol2tr{4,3,3,3,3}
or t\left\{\begin{array}{l}3, 3, 3\\3, 4\end{array}\right\}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node}}
{{CDD|node_1|split1|nodes_11|4a3b|nodes|3b|nodeb}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges
bgcolor=#e7dcc3|Vertices
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsB6, [3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Bicantitruncated hexeract
  • Great birhombihexeract (acronym: gaborx) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/gaborx.htm (o3o3x3x3x4o - gaborx)]}}

= Images =

{{6-cube Coxeter plane graphs|t123|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, [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}} o3o3o3x3o4x - srox, o3o3x3o3x4o - saborx, o3o3o3x3x4x - grox, o3o3x3x3x4o - gaborx {{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