Runcic 6-cubes

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File:6-demicube t0 D6.svg
6-demicube
File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:6-demicube t02 D6.svg
Runcic 6-cube
File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:6-demicube t012 D6.svg
Runcicantic 6-cube
File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
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

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Runcic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2{3,33,1}
h3{4,34}
Coxeter-Dynkin diagram File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces
4-faces
Cells
Faces
Edges 3840
Vertices 640
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

Alternate names

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

Cartesian coordinates

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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 | edit source]
orthographic projections
Coxeter plane B6
Graph File:6-demicube t02 B6.svg
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph File:6-demicube t02 D6.svg File:6-demicube t02 D5.svg
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph File:6-demicube t02 D4.svg File:6-demicube t02 D3.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph File:6-demicube t02 A5.svg File:6-demicube t02 A3.svg
Dihedral symmetry [6] [4]
[edit | edit source]
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 		File:Schlegel half-solid rectified 8-cell.png File:5-demicube t03 D5.svg File:6-demicube t03 D6.svg File:7-demicube t03 D7.svg File:8-demicube t03 D8.svg
Coxeter File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Schläfli h3{4,32} h3{4,33} h3{4,34} h3{4,35} h3{4,36}

Runcicantic 6-cube

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Runcicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2{3,33,1}
h2,3{4,34}
Coxeter-Dynkin diagram File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces
4-faces
Cells
Faces
Edges 5760
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

Alternate names

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

Cartesian coordinates

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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 | edit source]
orthographic projections
Coxeter plane B6
Graph File:6-demicube t012 B6.svg
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph File:6-demicube t012 D6.svg File:6-demicube t012 D5.svg
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph File:6-demicube t012 D4.svg File:6-demicube t012 D3.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph File:6-demicube t012 A5.svg File:6-demicube t012 A3.svg
Dihedral symmetry [6] [4]
[edit | edit source]

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
File:6-demicube t0 D6.svg
h{4,34} 		File:6-demicube t01 D6.svg
h2{4,34} 		File:6-demicube t02 D6.svg
h3{4,34} 		File:6-demicube t03 D6.svg
h4{4,34} 		File:6-demicube t04 D6.svg
h5{4,34} 		File:6-demicube t012 D6.svg
h2,3{4,34} 		File:6-demicube t013 D6.svg
h2,4{4,34} 		File:6-demicube t014 D6.svg
h2,5{4,34}
File:6-demicube t023 D6.svg
h3,4{4,34} 		File:6-demicube t024 D6.svg
h3,5{4,34} 		File:6-demicube t034 D6.svg
h4,5{4,34} 		File:6-demicube t0123 D6.svg
h2,3,4{4,34} 		File:6-demicube t0124 D6.svg
h2,3,5{4,34} 		File:6-demicube t0134 D6.svg
h2,4,5{4,34} 		File:6-demicube t0234 D6.svg
h3,4,5{4,34} 		File:6-demicube t01234 D6.svg
h2,3,4,5{4,34}

Notes

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

References

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  • 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, wiley.com, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
      • (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.
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). x3o3o *b3x3o3o, x3x3o *b3x3o3o
[edit | edit source]
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 • Polytope operations
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