Similarity (geometry) similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all rhombi would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all rectangles would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional. For given "n", all regular "n"-gons are
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