Runcinated 5-orthoplex
Encyclopedia

5-orthoplex

Runcinated 5-orthoplex

Runcinated 5-cube
Runcinated 5-cube
In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination of the regular 5-cube....



Runcitruncated 5-orthoplex

Runcicantellated 5-orthoplex

Runcicantitruncated 5-orthoplex

Runcitruncated 5-cube

Runcicantellated 5-cube

Runcicantitruncated 5-cube
Orthogonal projections in BC5 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 runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order 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 :...

 (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....

) of the regular 5-orthoplex.

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

Runcinated 5-orthoplex

Runcinated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,3{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...

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]
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 pentacross
  • Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)

Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of:
(0,1,1,1,2)

Runcitruncated 5-orthoplex

Runcitruncated 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 t0,1,3{3,3,3,4}
t0,1,3{3,31,1}
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...

s

4-faces 202
Cells 1560
Faces 3760
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]
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

  • Runcitruncated pentacross
  • Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutation
Permutation
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
(±3,±2,±1,±1,0)

Runcicantellated 5-orthoplex

Runcicantellated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3{3,3,3,4}
t0,2,3{3,3,31,1}
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...

colspan=2|
4-faces 202
Cells 1240
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]
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

  • Runcicantellated pentacross
  • Prismatorhombated triacontiditeron (Acronym: pirt) (Jonathan Bowers)

Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of:
(0,1,2,2,3)

Runcicantitruncated 5-orthoplex

Runcicantitruncated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{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...


4-faces 202
Cells 1560
Faces 4240
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...

s
BC5 [4,3,3,3]
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...

, 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 pentacross
  • Great prismated triacontiditeron (gippit) (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

This polytope is one 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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