Truncated 5-cube
Encyclopedia
5-cube |
Truncated 5-cube |
Bitruncated 5-cube |
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5-orthoplex |
Truncated 5-orthoplex Truncated 5-orthoplex In six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the... |
Bitruncated 5-orthoplex |
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| Orthogonal projections in BC5 Coxeter plane | |||
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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 truncated 5-cube is a convex uniform 5-polytope, being a truncation
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 5-cube.
There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the square faces of the 5-cube. The third and fourth truncations are more easily constructed as second and first truncations of the 5-orthoplex.
Truncated 5-cube
| Truncated 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{4,3,3,3} |
| 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 |
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| 4-faces | 42 |
| Cells | 200 |
| Faces | 400 |
| Edges | 400 |
| 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:... |
 Elongated tetrahedral pyramid |
| 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... |
Construction and coordinates
The truncated 5-cube may be constructed by truncatingTruncation (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 vertices of the 5-cube at
of the edge length. A regular 5-cell is formed at each truncated vertex.The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:
Images
The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.Bitruncated 5-cube
| Bitruncated 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 | t1,2{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 |
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| 4-faces | 42 |
| Cells | 280 |
| Faces | 720 |
| Edges | 800 |
| 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 |
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... |
Construction and coordinates
The bitruncated 5-cube may be constructed by bitruncatingBitruncation
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves....
the vertices of the 5-cube at
of the edge length.The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:
