Runcinated 5-demicube
Encyclopedia
5-cube |
Runcinated 5-demicube |
Runcitruncated 5-demicube |
Runcicantellated 5-demicube |
Runcicantitruncated 5-demicube |
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| Orthogonal projections in D5 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 runcinated 5-demicube is a convex uniform 5-polytope with a 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....
operation, a 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 :...
the uniform 5-demicube.
There are unique 4 runcinations of the 5-demicube, including permutations of truncations, and cantellations.
Runcinated 5-demicube
| Runcinated 5-demicube | |
|---|---|
| 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 | t0,3{3,32,1} |
| 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... |
|
| 4-faces | 82 |
| Cells | 480 |
| Faces | 720 |
| Edges | 400 |
| Vertices | 80 |
| 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 |
D5, [32,1,1] |
| 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 demipenteract
- Small prismated hemipenteract (siphin) (Jonathan Bowers)
Cartesian coordinates
The Cartesian coordinates for the 80 vertices of a runcinated demipenteract centered at the origin are coordinate permutations:- (±1,±1,±1,±1,±3)
with an odd number of plus signs.
Runcitruncated 5-demicube
| Runcitruncated 5-demicube | |
|---|---|
| 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 | t0,1,3{3,32,1} |
| 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... |
|
| 4-faces | 82 |
| Cells | 720 |
| Faces | 1840 |
| Edges | 1680 |
| 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 |
D5, [32,1,1] |
| 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 demipenteract centered at the origin are coordinate permutations:- (±1,±1,±3,±3,±5)
with an odd number of plus signs.
Runcicantellated 5-demicube
| Runcicantellated 5-demicube | |
|---|---|
| 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 | t0,2,3{3,32,1} |
| 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... |
|
| 4-faces | 82 |
| Cells | 560 |
| Faces | 1280 |
| Edges | 1120 |
| Vertices | 320 |
| 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 |
D5, [32,1,1] |
| 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 320 vertices of a runcicantellated demipenteract centered at the origin are coordinate permutations:- (±1,±1,±1,±3,±5)
with an odd number of plus signs.
Runcicantitruncated 5-demicube
| Runcicantitruncated 5-demicube | |
|---|---|
| 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 | t0,1,2,3{3,32,1} |
| Coxeter symbol | t0,1,2,3(121) |
| 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 | 82 |
| Cells | 720 |
| Faces | 2080 |
| Edges | 2400 |
| 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... s |
D5, [32,1,1] |
| 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 demipenteract centered at the origin are coordinate permutations:- (±1,±1,±3,±5,±7)
with an odd number of plus signs.
Related polytopes
This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopeUniform polytope
A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....
s called demihypercubes for being alternation of the hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
family.
There are 23 uniform polytera
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....
(uniform 5-polytope) that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.