5-orthoplex	Cantellated 5-orthoplex	Bicantellated 5-cube	Cantellated 5-cube Cantellated 5-cube In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.There are 6 unique cantellation for the 5-cube, including truncations...
5-cube	Cantitruncated 5-orthoplex	Bicantitruncated 5-cube	Cantitruncated 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 cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.

There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube.

Cantellated 5-orthoplex

Cantellated 5-orthoplex
Type	Uniform 5-polytope
Schläfli symbol	t0,2{3,3,3,4} t0,2{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	122
Cells	680
Faces	1520
Edges	1280
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

• Cantellated 5-orthoplex
• Bicantellated 5-demicube
• Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)

Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of:

(0,0,1,1,2)

Images

The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.

Cantitruncated 5-orthoplex

Cantitruncated 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	t0,1,2{3,3,3,4} t0,1,2{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	122
Cells	680
Faces	1520
Edges	1600
Vertices	640
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

• Cantitruncated pentacross
• Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate 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,±1,0,0)

Related polytopes

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

External links

• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary