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

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