Skew polygon Skew polygon In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The "interior" surface (or area) of such a polygon is not uniquely defined. Skew infinite polygons (apeirogons) have vertices which are not all collinear. A zig-zag skew polygon or antiprismatic polygon has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygon in 3 dimensions (and regular skew apeirogons in 2 dimensions) are always zig-zag. A regular skew polygon is isogonal with equal edge lengths. In 3 dimensions a regular skew
Regular polyhedron five have C×S symmetry but can only be realised with half the symmetry, that is C×A or icosahedral symmetry. They are all topologically equivalent to toroids. Their construction, by arranging "n" faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images. The Petrie dual of a regular polyhedron is a regular map whose vertices and edges correspond to the vertices and edges of the original polyhedron, and whose faces are the set of skew Petrie polygons. The usual nine