6-demicube
=
Runcic 6-cube
=
Runcicantic 6-cube
=
Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

Runcic 6-cube

Runcic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,2{3,33,1}
h3{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges3840
Vertices640
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names

  • Cantellated 6-demicube/demihexeract
  • Small rhombated hemihexeract (Acronym sirhax) (Jonathan Bowers)[1]

Cartesian coordinates

The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±3)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Runcic n-cubes
n45678
[1+,4,3n-2]
= [3,3n-3,1]		 [1+,4,32]
= [3,31,1]		 [1+,4,33]
= [3,32,1]		 [1+,4,34]
= [3,33,1]		 [1+,4,35]
= [3,34,1]		 [1+,4,36]
= [3,35,1]
Runcic
figure
Coxeter
=
=
=
=
=
Schläfli h3{4,32} h3{4,33} h3{4,34} h3{4,35} h3{4,36}

Runcicantic 6-cube

Runcicantic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,1,2{3,33,1}
h2,3{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges5760
Vertices1920
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names

  • Cantitruncated 6-demicube/demihexeract
  • Great rhombated hemihexeract (Acronym girhax) (Jonathan Bowers)[2]

Cartesian coordinates

The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±5,±5,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

D6 polytopes

h{4,34}
h2{4,34}
h3{4,34}
h4{4,34}
h5{4,34}
h2,3{4,34}
h2,4{4,34}
h2,5{4,34}

h3,4{4,34}
h3,5{4,34}
h4,5{4,34}
h2,3,4{4,34}
h2,3,5{4,34}
h2,4,5{4,34}
h3,4,5{4,34}
h2,3,4,5{4,34}

Notes

  1. ↑ Klitzing, (x3o3o *b3x3o3o - sirhax)
  2. ↑ Klitzing, (x3x3o *b3x3o3o - girhax)

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
      • (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. "6D uniform polytopes (polypeta)". x3o3o *b3x3o3o, x3x3o *b3x3o3o
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
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