Runcinated 5-cube
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5-cube

Runcinated 5-cube

Runcinated 5-orthoplex
Runcinated 5-orthoplex
In six-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation of the regular 5-orthoplex.There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations...



Runcitruncated 5-cube

Runcicantellated 5-cube

Runcicantitruncated 5-cube

Runcitruncated 5-orthoplex

Runcicantellated 5-orthoplex

Runcicantitruncated 5-orthoplex
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 runcinated 5-cube is a convex uniform 5-polytope that is 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....

 (a 3rd order truncation) of the regular 5-cube.

There are unique 8 degrees of runcinations of the 5-cube, along with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-orthoplex.

Runcinated 5-cube

Runcinated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,3{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 162
Cells 1200
Faces 2160
Edges 1440
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...

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...


Coordinates

The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:

Runcitruncated 5-cube

Runcitruncated 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 t0,1,3{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 162
Cells 1440
Faces 3680
Edges 3360
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
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

  • Runcitruncated penteract
  • Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:

Runcicantellated 5-cube

Runcicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,2,3{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 162
Cells 1200
Faces 2960
Edges 2880
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

  • Runcicantellated penteract
  • Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:

Runcicantitruncated 5-cube

Runcicantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{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 162
Cells 1440
Faces 4160
Edges 4800
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

  • Runcicantitruncated penteract
  • Biruncicantitruncated 16-cell / Biruncicantitruncated pentacross
  • great prismated penteract (gippin) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of an runcicantitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:

Related polytopes

These polytopes are a part of a set of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

External links

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
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