Cantic 6-cube
Cantic 6-cube Truncated 6-demicube | |
---|---|
D6 Coxeter plane projection | |
Type | uniform polypeton |
Schläfli symbol | t0,1{3,33,1} h2{4,34} |
Coxeter-Dynkin diagram | = |
5-faces | 76 |
4-faces | 636 |
Cells | 2080 |
Faces | 3200 |
Edges | 2160 |
Vertices | 480 |
Coxeter groups | D6, [33,1,1] |
Properties | convex |
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
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] |
Related polytopes
n | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|
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
- ↑ 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
External links
- Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
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 | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 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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