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Runcic 7-cubes

January 01, 1970


7-demicube
Runcic 7-cube
Runcicantic 7-cube

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.

Runcic 7-cube edit

Runcic 7-cube
Type uniform 7-polytope
Schläfli symbol t0,2{3,34,1}
h3{4,35}
Coxeter-Dynkin diagram            
             
5-faces
4-faces
Cells
Faces
Edges 16800
Vertices 2240
Vertex figure
Coxeter groups D7, [34,1,1]
Properties convex

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

Alternate names edit

  • Small rhombated hemihepteract (Acronym sirhesa) (Jonathan Bowers)[1]

Cartesian coordinates edit

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 edit

orthographic projections
Coxeter
plane 		B7 D7 D6
Graph      
Dihedral
symmetry 		[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph      
Dihedral
symmetry 		[8] [6] [4]
Coxeter
plane 		A5 A3
Graph    
Dihedral
symmetry 		[6] [4]

Runcicantic 7-cube edit

Runcicantic 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,2{3,34,1}
h2,3{4,35}
Coxeter-Dynkin diagram            
             
5-faces
4-faces
Cells
Faces
Edges 23520
Vertices 6720
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

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

Alternate names edit

  • Great rhombated hemihepteract (Acronym girhesa) (Jonathan Bowers)[2]

Cartesian coordinates edit

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 edit

orthographic projections
Coxeter
plane 		B7 D7 D6
Graph      
Dihedral
symmetry 		[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph      
Dihedral
symmetry 		[8] [6] [4]
Coxeter
plane 		A5 A3
Graph    
Dihedral
symmetry 		[6] [4]

Related polytopes edit

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:

Notes edit

  1. ^ Klitzing, (x3o3o *b3x3o3o3o - sirhesa)
  2. ^ Klitzing, (x3x3o *b3x3o3o3o - girhesa)

References edit

  • 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 [1]
      • (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.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o *b3x3o3o3o - sirhesa, x3x3o *b3x3o3o3o - girhesa

External links edit

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

January 01, 1970
runcic, cubes, demicube, runcic, cube, runcicantic, cube, orthogonal, projections, coxeter, plane, seven, dimensional, geometry, runcic, cube, convex, uniform, polytope, related, uniform, demicube, there, unique, forms, contents, runcic, cube, alternate, names. 7 demicube Runcic 7 cube Runcicantic 7 cube 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 Contents 1 Runcic 7 cube 1 1 Alternate names 1 2 Cartesian coordinates 1 3 Images 2 Runcicantic 7 cube 2 1 Alternate names 2 2 Cartesian coordinates 2 3 Images 3 Related polytopes 4 Notes 5 References 6 External linksRuncic 7 cube editRuncic 7 cube Type uniform 7 polytope Schlafli symbol t0 2 3 34 1 h3 4 35 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 4 faces Cells Faces Edges 16800 Vertices 2240 Vertex figure Coxeter groups D7 34 1 1 Properties convex A runcic 7 cube h3 4 35 has half the vertices of a runcinated 7 cube t0 3 4 35 Alternate names edit Small rhombated hemihepteract Acronym sirhesa Jonathan Bowers 1 Cartesian coordinates edit 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 edit orthographic projections Coxeterplane B7 D7 D6 Graph nbsp nbsp nbsp Dihedralsymmetry 14 2 12 10 Coxeter plane D5 D4 D3 Graph nbsp nbsp nbsp Dihedralsymmetry 8 6 4 Coxeterplane A5 A3 Graph nbsp nbsp Dihedralsymmetry 6 4 Runcicantic 7 cube editRuncicantic 7 cube Type uniform 7 polytope Schlafli symbol t0 1 2 3 34 1 h2 3 4 35 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 4 faces Cells Faces Edges 23520 Vertices 6720 Vertex figure Coxeter groups D6 33 1 1 Properties convex A runcicantic 7 cube h2 3 4 35 has half the vertices of a runcicantellated 7 cube t0 1 3 4 35 Alternate names edit Great rhombated hemihepteract Acronym girhesa Jonathan Bowers 2 Cartesian coordinates edit 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 edit orthographic projections Coxeterplane B7 D7 D6 Graph nbsp nbsp nbsp Dihedralsymmetry 14 2 12 10 Coxeter plane D5 D4 D3 Graph nbsp nbsp nbsp Dihedralsymmetry 8 6 4 Coxeterplane A5 A3 Graph nbsp nbsp Dihedralsymmetry 6 4 Related polytopes editThis 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 D7 polytopes nbsp t0 141 nbsp t0 1 141 nbsp t0 2 141 nbsp t0 3 141 nbsp t0 4 141 nbsp t0 5 141 nbsp t0 1 2 141 nbsp t0 1 3 141 nbsp t0 1 4 141 nbsp t0 1 5 141 nbsp t0 2 3 141 nbsp t0 2 4 141 nbsp t0 2 5 141 nbsp t0 3 4 141 nbsp t0 3 5 141 nbsp t0 4 5 141 nbsp t0 1 2 3 141 nbsp t0 1 2 4 141 nbsp t0 1 2 5 141 nbsp t0 1 3 4 141 nbsp t0 1 3 5 141 nbsp t0 1 4 5 141 nbsp t0 2 3 4 141 nbsp t0 2 3 5 141 nbsp t0 2 4 5 141 nbsp t0 3 4 5 141 nbsp t0 1 2 3 4 141 nbsp t0 1 2 3 5 141 nbsp t0 1 2 4 5 141 nbsp t0 1 3 4 5 141 nbsp t0 2 3 4 5 141 nbsp t0 1 2 3 4 5 141 Notes edit Klitzing x3o3o b3x3o3o3o sirhesa Klitzing x3x3o b3x3o3o3o girhesa References editH 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 1 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 Klitzing Richard 7D uniform polytopes polyexa x3o3o b3x3o3o3o sirhesa x3x3o b3x3o3o3o girhesaExternal links editWeisstein Eric W Hypercube MathWorld Polytopes of Various Dimensions Multi dimensional Glossary vteFundamental convex regular and uniform polytopes in dimensions 2 10 Family An Bn I2 p Dn E6 E7 E8 F4 G2 Hn Regular polygon Triangle Square p gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cell Uniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicube Uniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221 Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321 Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421 Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicube Uniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicube Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Runcic 7 cubes amp oldid 817103107 runcic 7 cube, wikipedia, wiki, book, books, library,

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