Runcinated 6-cube
Encyclopedia
6-cube |
Runcinated 6-cube |
Biruncinated 6-cube |
Runcinated 6-orthoplex Runcinated 6-orthoplex In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations of the regular 6-orthoplex.There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and cantellations... |
6-orthoplex |
Runcitruncated 6-cube |
Biruncitruncated 6-cube |
Runcicantellated 6-orthoplex |
Runcicantellated 6-cube |
Biruncitruncated 6-orthoplex |
Runcitruncated 6-orthoplex |
Runcicantitruncated 6-cube |
Biruncicantitruncated 6-cube |
Runcicantitruncated 6-orthoplex |
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| Orthogonal projections in BC6 Coxeter plane | ||||
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In six-dimensional geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...
(runcination
Runcination
In geometry, runcination is an operation that cuts a regular polytope simultaneously along the faces, edges and vertices, creating new facets in place of the original face, edge, and vertex centers....
) of the regular 6-cube.
There are 12 unique runcinations of the 6-cube with permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s of truncations, and cantellations. Half are expressed relative to the dual 6-orthoplex.
Runcinated 6-cube
| Runcinated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7680 |
| Vertices | 1280 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
BC6 [4,3,3,3,3] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Biruncinated 6-cube
| Biruncinated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 11520 |
| Vertices | 1920 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
BC6 [4,3,3,3,3] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Runcitruncated 6-cube
| Runcitruncated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,1,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 17280 |
| Vertices | 3840 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
BC6 [4,3,3,3,3] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Biruncitruncated 6-cube
| Biruncitruncated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,2,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 5760 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
BC6 [4,3,3,3,3] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Runcicantellated 6-cube
| Runcicantellated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,2,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3840 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
BC6 [4,3,3,3,3] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Runcicantitruncated 6-cube
| Runcicantitruncated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,1,2,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 7680 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
BC6 [4,3,3,3,3] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Biruncitruncated 6-cube
| Biruncitruncated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,2,3,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
|
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 5760 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
|
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
BC6 [4,3,3,3,3] |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |