Cantellated 5-cell


In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation of the regular 5-cell.
There are 2 unique degrees of runcinations of the 5-cell including with permutations truncations.

Cantellated 5-cell

The cantellated 5-cell or small rhombated pentachroron is a uniform 4-polytope. It has 30 vertices, 90 edges, 80 faces, and 20 cells. The cells are 5 cuboctahedra, 5 octahedra, and 10 triangular prisms. Each vertex is surrounded by 2 cuboctahedra, 2 triangular prisms, and 1 octahedron; the vertex figure is a nonuniform triangular prism.

Alternate names


Wireframe

Ten triangular prisms colored green

Five octahedra colored blue

Coordinates

The Cartesian coordinates of the vertices of the origin-centered cantellated 5-cell having edge length 2 are:
The vertices of the cantellated 5-cell can be most simply positioned in 5-space as permutations of:
This construction is from the positive orthant facet of the cantellated 5-orthoplex.

Related polytopes

The convex hull of two cantellated 5-cells in opposite positions is a nonuniform polychoron composed of 100 cells: three kinds of 70 octahedra, 30 tetrahedra, and 60 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.

Vertex figure'''

Cantitruncated 5-cell

The cantitruncated 5-cell or great rhombated pentachoron is a uniform 4-polytope. It is composed of 60 vertices, 120 edges, 80 faces, and 20 cells. The cells are: 5 truncated octahedra, 10 triangular prisms, and 5 truncated tetrahedra. Each vertex is surrounded by 2 truncated octahedra, one triangular prism, and one truncated tetrahedron.

Alternative names


Stereographic projection with its 10 triangular prisms.

Cartesian coordinates

The Cartesian coordinates of an origin-centered cantitruncated 5-cell having edge length 2 are:
These vertices can be more simply constructed on a hyperplane in 5-space, as the permutations of:
This construction is from the positive orthant facet of the cantitruncated 5-orthoplex.

Related polytopes

A double symmetry construction can be made by placing truncated tetrahedra on the truncated octahedra, resulting in a nonuniform polychoron with 10 truncated tetrahedra, 20 hexagonal prisms, two kinds of 80 triangular prisms, and 30 tetrahedra. Its vertex figure is topologically equivalent to the octahedron.

Vertex figure'''

Related 4-polytopes

These polytopes are art of a set of 9 Uniform 4-polytopes constructed from the Coxeter group.