In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.[1]
Infinite regular skew polyhedra that span 3-space or higher are called regular skew apeirohedra.
According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra.
Coxeter offered a modified Schläfli symbol for these figures, with implying the vertex figure, -gons around a vertex, and -gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.
The regular skew polyhedra, represented by, follow this equation:
2\cos
| \pi | |
| l |
\cos
| \pi | |
| m |
=\cos
| \pi | |
| n |
A first set, repeats the five convex Platonic solids, and one nonconvex Kepler–Poinsot solid: