Regular polygon Regular polygon In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. "These properties apply to all regular polygons, whether convex or star." A regular "n"-sided polygon has rotational symmetry of order "n". All vertices of a regular
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