Stericated 5-cube
Encyclopedia
5-cube |
Stericated 5-cube |
Steritruncated 5-cube |
Stericantellated 5-cube |
Steritruncated 5-orthoplex |
Stericantitruncated 5-cube |
Steriruncitruncated 5-cube |
Stericantitruncated 5-orthoplex |
Omnitruncated 5-cube |
| Orthogonal projections in BC5 Coxeter plane | ||
|---|---|---|
In five-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 stericated 5-cube is a convex uniform 5-polytope with fourth-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 :...
(sterication) of the regular 5-cube.
There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible
Constructibility
In mathematics, there are several notions of constructibility. Each of the following is by definition constructible:* a point in the Euclidean plane that can be constructed with compass and straightedge...
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 5-cube. The highest form, the steriruncicantitruncated 5-cube, is more simply called a omnitruncated 5-cube with all of the nodes ringed.
Stericated 5-cube
| Stericated 5-cube | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,4{4,3,3,3} | |
| 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... |
colspan=2| | |
| 4-faces | 242 | |
| Cells | 800 | |
| Faces | 1040 | |
| Edges | 640 | |
| Vertices | 160 | |
| 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... |
BC5 [4,3,3,3] | |
| 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
- Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
- Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
- Small cellated penteract (Acronym: scan) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a stericated 5-cube having edge length 2 are all permutations of:
Images
The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.Steritruncated 5-cube
| Steritruncated 5-cube | |
|---|---|
| Type | uniform polyteron Uniform polyteron In geometry, a uniform polyteron is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive and constructed from uniform polychoron facets.... |
| Schläfli symbol | t0,1,4{4,3,3,3} |
| 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 |
|
| 4-faces | 242 |
| Cells | 1520 |
| Faces | 2880 |
| Edges | 2240 |
| Vertices | 640 |
| 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 |
BC5, [3,3,3,4] |
| 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
- Steritruncated penteract
- Prismatotruncated penteract (Acronym: capt) (Jonathan Bowers)
Construction and coordinates
The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:
Stericantellated 5-cube
| Stericantellated 5-cube | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,2,4{4,3,3,3} | |
| 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... |
colspan=2| | |
| 4-faces | 242 | |
| Cells | 2080 | |
| Faces | 4720 | |
| Edges | 3840 | |
| Vertices | 960 | |
| 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... |
BC5 [4,3,3,3] | |
| 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
- Stericantellated penteract
- Stericantellated 5-orthoplex, stericantellated pentacross
- Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a stericantellated 5-cube having edge length 2 are all permutations of:
Stericantitruncated 5-cube
| Stericantitruncated 5-cube | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,1,2,4{4,3,3,3} | |
| 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 | 242 | |
| Cells | 2320 | |
| Faces | 5920 | |
| Edges | 5760 | |
| 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... |
BC5 [4,3,3,3] | |
| 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 |
|
Alternate names
- Stericantitruncated penteract
- Steriruncicantellated 16-cell / Biruncicantitruncated pentacross
- Celligreatorhombated penteract (cogrin) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of an stericantitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:
Steriruncitruncated 5-cube
| Steriruncitruncated 5-cube | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,1,3,4{4,3,3,3} | |
| 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 | 242 | |
| Cells | 2160 | |
| Faces | 5760 | |
| Edges | 5760 | |
| 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... |
BC5 [4,3,3,3] | |
| 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.... |
|
Alternate names
- Steriruncitruncated penteract
- Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of an steriruncitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:
Steritruncated 5-orthoplex
| Steritruncated 5-orthoplex | |
|---|---|
| Type | uniform polyteron Uniform polyteron In geometry, a uniform polyteron is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive and constructed from uniform polychoron facets.... |
| Schläfli symbol | t0,1,4{3,3,3,4} |
| 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 |
|
| 4-faces | 242 |
| Cells | 1600 |
| Faces | 2960 |
| Edges | 2240 |
| Vertices | 640 |
| 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... |
BC5, [3,3,3,4] |
| 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
- Steritruncated pentacross
- Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a Steritruncated 5-orthoplex, centered at the origin, are all permutationPermutation
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
Stericantitruncated 5-orthoplex
| Stericantitruncated 5-orthoplex | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,1,2,4{4,3,3,3} | |
| 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 | 242 | |
| Cells | 2400 | |
| Faces | 6000 | |
| Edges | 5760 | |
| 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... |
BC5 [4,3,3,3] | |
| 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.... |
|
Alternate names
- Stericantitruncated pentacross
- Celligreatorhombated pentacross (cogart) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of an stericantitruncated 5-orthoplex having an edge length of 2 are given by all permutations of coordinates and sign of:
Omnitruncated 5-cube
| Omnitruncated 5-cube | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,1,2,3,4{4,3,3,3} | |
| 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 | 242 | |
| Cells | 2640 | |
| Faces | 8160 | |
| Edges | 9600 | |
| 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:... |
 irr. {3,3,3} |
|
| 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... |
BC5 [4,3,3,3] | |
| 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.... |
|
Alternate names
- Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
- Omnitruncated penteract
- Omnitruncated 16-cell / omnitruncated pentacross
- Great cellated penteractitriacontiditeron (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of an omnitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:
Related polytopes
This polytope is one of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.External links
- Polytopes of Various Dimensions, Jonathan Bowers
- Multi-dimensional Glossary