Steric 6-cubes

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6-demicube t0 D5.svg
6-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-demicube t03 D5.svg
Steric 6-cube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-demicube t013 D5.svg
Stericantic 6-cube
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-demicube t023 D5.svg
Steriruncic 6-cube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-demicube t0123 D5.svg
Stericruncicantic 6-cube
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Orthogonal projections in D6 Coxeter plane

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

Steric 6-cube[edit]

Steric 6-cube
Type uniform 6-polytope
Schläfli symbol t0,3{3,33,1}
h4{4,34}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 480
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

Alternate names[edit]

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

Cartesian coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B6
Graph 6-demicube t03 B6.svg
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph 6-demicube t03 D6.svg 6-demicube t03 D5.svg
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph 6-demicube t03 D4.svg 6-demicube t03 D3.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-demicube t03 A5.svg 6-demicube t03 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

Stericantic 6-cube[edit]

Stericantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,3{3,33,1}
h2,4{4,34}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-faces
4-faces
Cells
Faces
Edges 12960
Vertices 2880
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

Alternate names[edit]

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

Cartesian coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B6
Graph 6-demicube t013 B6.svg
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph 6-demicube t013 D6.svg 6-demicube t013 D5.svg
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph 6-demicube t013 D4.svg 6-demicube t013 D3.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-demicube t013 A5.svg 6-demicube t013 A3.svg
Dihedral symmetry [6] [4]

Steriruncic 6-cube[edit]

Steriruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,3{3,33,1}
h3,4{4,34}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-faces
4-faces
Cells
Faces
Edges 7680
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

Alternate names[edit]

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

Cartesian coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B6
Graph 6-demicube t023 B6.svg
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph 6-demicube t023 D6.svg 6-demicube t023 D5.svg
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph 6-demicube t023 D4.svg 6-demicube t023 D3.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-demicube t023 A5.svg 6-demicube t023 A3.svg
Dihedral symmetry [6] [4]

Steriruncicantic 6-cube[edit]

Steriruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,3{3,32,1}
h2,3,4{4,34}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-faces
4-faces
Cells
Faces
Edges 17280
Vertices 5760
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

Alternate names[edit]

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

Cartesian coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B6
Graph 6-demicube t0123 B6.svg
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph 6-demicube t0123 D6.svg 6-demicube t0123 D5.svg
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph 6-demicube t0123 D4.svg 6-demicube t0123 D3.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-demicube t0123 A5.svg 6-demicube t0123 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

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

Notes[edit]

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

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 *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax

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