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

• Polytopes of Various Dimensions, Jonathan Bowers
  • Runcinated uniform polytera (spid), Jonathan Bowers
• Multi-dimensional Glossary