Runcinated 5-cell

Topologically, under its highest symmetry, [[3,3,3]], there is only one geometrical form, containing 10 tetrahedra and 20 uniform triangular prisms.

A second construction in 5-space, from the center of a rectified 5-orthoplex is given by coordinate permutations of: Its 20 vertices represent the root vectors of the simple Lie group A4.

This cross-section divides the runcinated 5-cell into two tetrahedral hypercupolae consisting of 5 tetrahedra and 10 triangular prisms each.

The dual regular skew polyhedron, {6,4|3}, is similarly related to the hexagonal faces of the bitruncated 5-cell.

The runcitruncated 5-cell or prismatorhombated pentachoron is composed of 60 vertices, 150 edges, 120 faces, and 30 cells.

The omnitruncated 5-cell or great prismatodecachoron is composed of 120 vertices, 240 edges, 150 faces (90 squares and 60 hexagons), and 30 cells.

The omnitruncated 5-cell honeycomb can tessellate 4-dimensional space by translational copies of this cell, each with 3 hypercells around each face.

The vertex figure of the omnitruncated 5-cell represents the Goursat tetrahedron of the [3,3,3] Coxeter group.

Nonuniform variants with [3,3,3] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra on each other to produce a nonuniform polychoron with 10 truncated octahedra, two types of 40 hexagonal prisms (20 ditrigonal prisms and 20 ditrigonal trapezoprisms), two kinds of 90 rectangular trapezoprisms (30 with D2d symmetry and 60 with C2v symmetry), and 240 vertices.

Vertex figure This polychoron can then be alternated to produce another nonuniform polychoron with 10 icosahedra, two types of 40 octahedra (20 with S6 symmetry and 20 with D3 symmetry), three kinds of 210 tetrahedra (30 tetragonal disphenoids, 60 phyllic disphenoids, and 120 irregular tetrahedra), and 120 vertices.

Vertex figure The full snub 5-cell or omnisnub 5-cell, defined as an alternation of the omnitruncated 5-cell, cannot be made uniform, but it can be given Coxeter diagram , and symmetry [[3,3,3]]+, order 120, and constructed from 90 cells: 10 icosahedrons, 20 octahedrons, and 60 tetrahedrons filling the gaps at the deleted vertices.