Runcic 7-cubes

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7-demicube t0 D7.svg
7-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-demicube t02 D7.svg
Runcic 7-cube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-demicube t012 D7.svg
Runcicantic 7-cube
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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 7-demicube t02 B7.svg 7-demicube t02 D7.svg 7-demicube t02 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t02 D5.svg 7-demicube t02 D4.svg 7-demicube t02 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t02 A5.svg 7-demicube t02 A3.svg
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 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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 7-demicube t012 B7.svg 7-demicube t012 D7.svg 7-demicube t012 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t012 D5.svg 7-demicube t012 D4.svg 7-demicube t012 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t012 A5.svg 7-demicube t012 A3.svg
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 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds