| Cantic 6-cube Truncated 6-demicube | |
|---|---|
 D6 Coxeter plane projection | |
| Type | uniform polypeton |
| Schläfli symbol | t0,1{3,3} h2{4,3} |
| Coxeter-Dynkin diagram |      =  
|
| 5-faces | 76 |
| 4-faces | 636 |
| Cells | 2080 |
| Faces | 3200 |
| Edges | 2160 |
| Vertices | 480 |
| Vertex figure | ( )v |
| Coxeter groups | D6, |
| 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)
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 | ||
| Coxeter plane | D6 | D5 |
| Graph | 
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| Dihedral symmetry | ||
| Coxeter plane | D4 | D3 |
| Graph | 
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| Dihedral symmetry | ||
| Coxeter plane | A5 | A3 |
| Graph | 
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| Dihedral symmetry |
Related polytopes
| n | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| Symmetry |
= |
= |
= |
= |
= |
= |
| Cantic figure |
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| Coxeter | =
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=
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=
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=
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=
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=
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| Schläfli | h2{4,3} | h2{4,3} | h2{4,3} | h2{4,3} | h2{4,3} | h2{4,3} |
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
| D6 polytopes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
 h{4,3} |
h2{4,3} |
 h3{4,3} |
 h4{4,3} |
 h5{4,3} |
 h2,3{4,3} |
 h2,4{4,3} |
 h2,5{4,3} | ||||
 h3,4{4,3} |
 h3,5{4,3} |
 h4,5{4,3} |
 h2,3,4{4,3} |
 h2,3,5{4,3} |
 h2,4,5{4,3} |
 h3,4,5{4,3} |
 h2,3,4,5{4,3} | ||||
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,
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
- 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
| 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 | ||||||||||||