![]() 6-demicube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcic 6-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcicantic 6-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
Orthogonal projections in D6 Coxeter plane |
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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/demihexeract
- Small rhombated hemihexeract (Acronym sirhax) (Jonathan Bowers)[1]
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 | |||||||||||
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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 |
![]() |
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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/demihexeract
- Great rhombated hemihexeract (Acronym girhax) (Jonathan Bowers)[2]
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
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 [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 *b3x3o3o, x3x3o *b3x3o3o
External links
- Weisstein, Eric W. "Hypercube". MathWorld.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary