Cantic 6-cube
Truncated 6-demicube

D6 Coxeter plane projection
Typeuniform polypeton
Schläfli symbolt0,1{3,33,1}
h2{4,34}
Coxeter-Dynkin diagram =
5-faces76
4-faces636
Cells2080
Faces3200
Edges2160
Vertices480
Vertex figure( )v[{ }x{3,3}]
Coxeter groupsD6, [33,1,1]
Propertiesconvex

In six-dimensional geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope.

Alternate names

  • Truncated 6-demicube/demihexeract (Acronym thax) (Jonathan Bowers)[1]

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a cantic 6-cube centered at the origin and edge length 6√2 are coordinate permutations:

(±1,±1,±3,±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]
Dimensional family of cantic n-cubes
n345678
Symmetry
[1+,4,3n-2]		 [1+,4,3]
= [3,3]		 [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]
Cantic
figure
Coxeter
=
=
=
=
=
=
Schläfli h2{4,3} h2{4,32} h2{4,33} h2{4,34} h2{4,35} h2{4,36}

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. ↑ Klitizing, (x3x3o *b3o3o3o – thax)

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)". x3x3o *b3o3o3o – thax
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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