Steric 6-cubes

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.


6-demicube
=
Steric 6-cube
=
Stericantic 6-cube
=

Steriruncic 6-cube
=
Stericruncicantic 6-cube
=
Orthogonal projections in D6 Coxeter plane

Steric 6-cube
Steric 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,3{3,33,1}
h4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges3360
Vertices480
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names
  • Runcinated demihexeract/6-demicube
  • Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers)[1]

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±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 steric n-cubes
n5678
[1+,4,3n-2]
= [3,3n-3,1]		 [1+,4,33]
= [3,32,1]		 [1+,4,34]
= [3,33,1]		 [1+,4,35]
= [3,34,1]		 [1+,4,36]
= [3,35,1]
Steric
figure
Coxeter
=
=
=
=
Schläfli h4{4,33} h4{4,34} h4{4,35} h4{4,36}

Stericantic 6-cube
Stericantic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,1,3{3,33,1}
h2,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges12960
Vertices2880
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names
  • Runcitruncated demihexeract/6-demicube
  • Prismatotruncated hemihexeract (Acronym pithax) (Jonathan Bowers)[2]

Cartesian coordinates

The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±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]

Steriruncic 6-cube
Steriruncic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,2,3{3,33,1}
h3,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges7680
Vertices1920
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names
  • Runcicantellated demihexeract/6-demicube
  • Prismatorhombated hemihexeract (Acronym prohax) (Jonathan Bowers)[3]

Cartesian coordinates

The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±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]

Steriruncicantic 6-cube
Steriruncicantic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,1,2,3{3,32,1}
h2,3,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges17280
Vertices5760
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names
  • Runcicantitruncated demihexeract/6-demicube
  • Great prismated hemihexeract (Acronym gophax) (Jonathan Bowers)[4]

Cartesian coordinates

The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

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]

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 *b3o3x3o - sophax)
  2. Klitzing, (x3x3o *b3o3x3o - pithax)
  3. Klitzing, (x3o3o *b3x3x3o - prohax)
  4. Klitzing, (x3x3o *b3x3x3o - gophax)

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 *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax
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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.