Runcinated 6-demicube
Encyclopedia
6-cube |
Runcinated 6-demicube |
Runcitruncated 6-demicube |
Runcicantellated 6-demicube |
Runcicantitruncated 6-demicube |
|
| Orthogonal projections in D6 Coxeter plane | ||
|---|---|---|
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-demicube 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 uniform 6-demicube.
There are unique 4 runcinations of the 6-demicube, including permutations of truncations, and cantellations.
Runcinated 6-demicube
| Runcinated 6-demicube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,3{3,33,1} |
| 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... |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3360 |
| Vertices | 480 |
| 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... s |
D6, [33,1,1] |
| 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... |
Alternate names
- runcinated demihexeract
- Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers)
Cartesian coordinates
The Cartesian coordinates for the vertices of a runcinated demihexeract centered at the origin are coordinate permutations:- (±1,±1,±1,±1,±1,±3)
with an odd number of plus signs.
Runcitruncated 6-demicube
| Runcitruncated 6-demicube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,1,3{3,33,1} |
| 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... |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 12960 |
| Vertices | 2880 |
| 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... s |
D6, [33,1,1] |
| 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... |
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a runcicantitruncated demihexeract centered at the origin are coordinate permutations:- (±1,±1,±1,±3,±3,±5)
with an odd number of plus signs.
Runcicantellated 6-demicube
| Runcicantellated 6-demicube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,2,3{3,33,1} |
| 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... |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7680 |
| 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... s |
D6, [33,1,1] |
| 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... |
Cartesian coordinates
The Cartesian coordinates for the vertices of a runcicantellated demihexeract centered at the origin are coordinate permutations:- (±1,±1,±1,±1,±3,±5)
with an odd number of plus signs.
Runcicantitruncated 6-demicube
| Runcicantitruncated 6-demicube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,1,2,3{3,32,1} |
| Coxeter symbol | t0,1,2,3(131) |
| 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... |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 17280 |
| 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... s |
D6, [33,1,1] |
| 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... |
Cartesian coordinates
The Cartesian coordinates for the 960 vertices of a runcicantitruncated demihexeract centered at the origin are coordinate permutations:- (±1,±1,±1,±3,±5,±7)
with an odd number of plus signs.