Pentellated 6-cube
Encyclopedia
6-cube |
6-orthoplex |
Pentellated 6-cube |
Pentitruncated 6-cube |
Penticantellated 6-cube |
Penticantitruncated 6-cube |
Pentiruncitruncated 6-cube |
Pentiruncicantellated 6-cube |
Pentiruncicantitruncated 6-cube |
Pentisteritruncated 6-cube |
Pentistericantitruncated 6-cube |
Omnitruncated 6-cube |
| Orthogonal projections in BC6 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 pentellated 6-cube is a convex uniform 6-polytope with 5th 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 :...
of the regular 6-cube.
There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...
operation applied to the regular 6-cube. The highest form, the pentisteriruncicantitruncated 6-cube, is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex
Pentellated 6-orthoplex
In six-dimensional geometry, a pentellated 6-orthoplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-orthoplex.There are unique 16 degrees of pentellations of the 6-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. Ten are shown,...
.
Pentellated 6-cube
| Pentellated 6-cube | |
|---|---|
| Type | Uniform polypeton |
| Schläfli symbol | t0,5{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... |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1920 |
| Vertices | 384 |
| 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:... |
5-cell antiprism |
| 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... |
Alternate names
- Pentellated 6-orthoplex
- Expanded 6-cube, expanded 6-orthoplex
- Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers)
Pentitruncated 6-cube
| Pentitruncated 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,1,5{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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8640 |
| 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 |
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... |
Penticantellated 6-cube
| Penticantellated 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,2,5{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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 21120 |
| 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... s |
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... |
Penticantitruncated 6-cube
| Penticantitruncated 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,1,2,5{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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30720 |
| 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... s |
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... |
Pentiruncitruncated 6-cube
| Pentiruncitruncated 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,1,3,5{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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 151840 |
| Vertices | 11520 |
| 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 |
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... |
Pentiruncicantellated 6-cube
| Pentiruncicantellated 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,2,3,5{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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 46080 |
| Vertices | 11520 |
| 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 |
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... |
Alternate names
- Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers)
Pentiruncicantitruncated 6-cube
| Pentiruncicantitruncated 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,1,2,3,5{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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 23040 |
| 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 |
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... |
Pentisteritruncated 6-cube
| Pentisteritruncated 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,1,4,5{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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30720 |
| 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... s |
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... |
Alternate names
- Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers)
Pentistericantitruncated 6-cube
| Pentistericantitruncated 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t0,1,2,4,5{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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 23040 |
| 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 |
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... |
Omnitruncated 6-cube
| Omnitruncated 6-cube | |
|---|---|
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,1,2,3,4,5{35} |
| 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... s |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 138240 |
| Vertices | 46080 |
| 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:... |
irregular 5-simplex |
| 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... , isogonal Isogonal Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts:*Isogonal polygon, polyhedron, polytope or tiling.*Isogonal trajectory in curve theory.*Isogonal conjugate in triangle geometry.... |
The omnitruncated 6-cube has 5040 vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
, 15120 edge
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
s, 16800 faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
(4200 hexagons and 1260 squares
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube.
Alternate names
- Pentisteriruncicantituncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes)
- Omnitruncated hexeract
- Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers)