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Runcic 7-cubes#runcic 7-cube

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|160px
7-demicube
{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}}
{{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node}}

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Runcic 7-cube
{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node}}
{{CDD|node_h|4|node|3|node|3|node_1|3|node|3|node|3|node}}

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Runcicantic 7-cube
{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node}}
{{CDD|node_h|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}

colspan=4|Orthogonal projections in D7 Coxeter plane

In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

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Runcic 7-cube

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!bgcolor=#e7dcc3 colspan=2|Runcic 7-cube

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,2{3,34,1}
h3{4,35}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node}}
{{CDD|node_h|4|node|3|node|3|node_1|3|node|3|node|3|node}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges16800
bgcolor=#e7dcc3|Vertices2240
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD7, [34,1,1]
bgcolor=#e7dcc3|Propertiesconvex

A runcic 7-cube, h3{4,35}, has half the vertices of a runcinated 7-cube, t0,3{4,35}.

=Alternate names=

  • Small rhombated hemihepteract (Acronym sirhesa) (Jonathan Bowers)Klitzing, (x3o3o *b3x3o3o3o - sirhesa)

=Cartesian coordinates=

The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations:

: (±1,±1,±1,±3,±3,±3,±3)

with an odd number of plus signs.

=Images=

{{7-demicube Coxeter plane graphs|t02|150}}

Runcicantic 7-cube

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!bgcolor=#e7dcc3 colspan=2|Runcicantic 7-cube

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,2{3,34,1}
h2,3{4,35}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node}}
{{CDD|node_h|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges23520
bgcolor=#e7dcc3|Vertices6720
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

A runcicantic 7-cube, h2,3{4,35}, has half the vertices of a runcicantellated 7-cube, t0,1,3{4,35}.

= Alternate names=

  • Great rhombated hemihepteract (Acronym girhesa) (Jonathan Bowers)Klitzing, (x3x3o *b3x3o3o3o - girhesa)

=Cartesian coordinates=

The Cartesian coordinates for the vertices of a runcicantic 7-cube centered at the origin are coordinate permutations:

: (±1,±1,±1,±1,±3,±5,±5)

with an odd number of plus signs.

=Images=

{{7-demicube Coxeter plane graphs|t012|150}}

Related polytopes

This polytope is based on the 7-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC6 symmetry, and 32 are unique:

{{Demihepteract_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, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • (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|polyexa.htm|7D|uniform polytopes (polyexa)}} x3o3o *b3x3o3o3o - sirhesa, x3x3o *b3x3o3o3o - girhesa

External links

  • {{MathWorld|title=Hypercube|urlname=Hypercube}}
  • [http://www.polytopes.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:7-polytopes