Skew polytope

A regular skew polytope is an n-dimensional regular polytope where the vertices do not all line in the same n-dimensional space.

Skew polygons
Consider a uniform antiprism polyhedron. The edges can be divided into three sets, viz. the edges of the two capping polygons and the zigzagging edges that form a ring around the middle. This last set forms a polygon that is both vertex regular and edge regular, making it a regular polygon. The vertices lie in two planes, as opposed to the one in which the more conventional polygons exist.

Petrie polygons
In the above description, the distance between alternating vertices is the edge length, but this need not be so. In particular, every regular polytope (of 3 or more dimensions) has a corresponding skew polygon. This is formed by taking connecting edges of the polytope, with the following rule: every n – 1 connected edges are in the same facet, but n connected edges do not share a facet.

These are the Petrie polygons.

Skew polyhedra
The five Platonic solids, namely, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron have been known for thousands of years, and that these were the only convex regular polyhedra. Johannes Kepler discovered the great icosahedron and the great stellated dodecahedron. There were two other polyhedra that Kepler overlooked (as they don&rsquo;t obey the Euler equality, $$F + V = E + 2$$), and it was left to Louis Poinsot to discover the small stellated dodecahedron and the great dodecahedron.

This was proved to be the complete set by Augustin-Louis Cauchy.

However, to quote Coxeter,

"One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedra; infinite, but free from false vertices . When my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex [fig], and one consisting of hexagons, four at each vertex [fig]. It was useless to protest that there is no room for more than four squares round a vertex.  The trick is, to let the faces go up and down in a kind of zig-zag formation, so that the faces that adjoin a given &ldquo;horizontal&rdquo; face lie alternately &ldquo;above&rdquo; and &ldquo;below&rdquo; it.  When I understood this, I pointed out a third possibility: hexagons, six at each vertex [fig]."