Regular skew polyhedron

The regular skew polyhedra, represented by {l,m|n}, follow this equation: A first set {l,m|n}, repeats the five convex Platonic solids, and one nonconvex Kepler–Poinsot solid: Coxeter also enumerated the a larger set of finite regular polyhedra in his paper "regular skew polyhedra in three and four dimensions, and their topological analogues".

Just like the infinite skew polyhedra represent manifold surfaces between the cells of the convex uniform honeycombs, the finite forms all represent manifold surfaces within the cells of the uniform 4-polytopes.

Polyhedra of the form {2p, 2q | r} are related to Coxeter group symmetry of [(p,r,q,r)], which reduces to the linear [r,p,r] when q is 2.

{4,4|n} produces a n-n duoprism, and specifically {4,4|4} fits inside of a {4}x{4} tesseract.

A final set is based on Coxeter's further extended form {q1,m|q2,q3...} or with q2 unspecified: {l, m |, q}.

A ring of 60 triangles make a regular skew polyhedron within a subset of faces of a 600-cell .
{4,5| 4} can be realized within the 32 vertices and 80 edges of a 5-cube , seen here in B5 Coxeter plane projection showing vertices and edges. The 80 square faces of the 5-cube become 40 square faces of the skew polyhedron and 40 square holes.