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Runcic 6-cubes

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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

[edit]
Runcic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2{3,33,1}
h3{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 3840
Vertices 640
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

Alternate names

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

Cartesian coordinates

[edit]

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

[edit]
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]
[edit]
Runcic n-cubes
n 4 5 6 7 8
[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

[edit]
Runcicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2{3,33,1}
h2,3{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 5760
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

Alternate names

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

Cartesian coordinates

[edit]

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

[edit]
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]
[edit]

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

[edit]
  1. ^ Klitzing, (x3o3o *b3x3o3o - sirhax)
  2. ^ Klitzing, (x3x3o *b3x3o3o - girhax)

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. "6D uniform polytopes (polypeta)". x3o3o *b3x3o3o, x3x3o *b3x3o3o
[edit]
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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