Runcic 6-cubes
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| Orthogonal projections in D6 Coxeter plane | ||
|---|---|---|
| 6-demicube = | Runcic 6-cube = | Runcicantic 6-cube = |
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
| Runcic 6-cube | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,2{3,33,1} h3{4,34} |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3840 |
| Vertices | 640 |
| Vertex figure | |
| Coxeter groups | D6, [33,1,1] |
| Properties | convex |
Alternate names
- Cantellated 6-demicube
- Cantellated demihexeract
- Small rhombated hemihexeract (Acronym: sirhax) (Jonathan Bowers)
Cartesian coordinates
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
| 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
| 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 | |||||
| Coxeter | = | = | = | = | = |
| Schläfli | h3{4,32} | h3{4,33} | h3{4,34} | h3{4,35} | h3{4,36} |
Runcicantic 6-cube
| Runcicantic 6-cube | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,1,2{3,33,1} h2,3{4,34} |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5760 |
| Vertices | 1920 |
| Vertex figure | |
| Coxeter groups | D6, [33,1,1] |
| Properties | convex |
Alternate names
- Cantitruncated 6-demicube
- Cantitruncated demihexeract
- Great rhombated hemihexeract (Acronym: girhax) (Jonathan Bowers)
Cartesian coordinates
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
| 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
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 | |||||||
|---|---|---|---|---|---|---|---|
| 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
- 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 Ivić 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. . x3o3o *b3x3o3o, x3x3o *b3x3o3o
External links
| vteFundamental 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 |