It seems natural to define a regular polygon as one whose sides are all the same length and whose angles between consecutive sides are all equal. A square, for example, is a regular polygon, whereas a rectangle (other than a square) is not. Similarly, a rhombus (other than a square) is not a regular polygon either. Using modern terminology, this general definition can be split into two, highlighting different types of regularity.
A polygon is said to be isogonal if any two vertices can be mapped to each other by an isometry of the polygon. It follows that the angle between any two consecutive sides is always the same. A rectangle is an example of an isogonal polygon, since all its angles are right angles.
A polygon is said to be isotoxal (from the Greek toxou, "arc") if any two sides can be mapped to each other by an isometry of the polygon. Thus, all its sides have the same length. A rhombus is an example of an isotoxal polygon.
Isotoxal polygons are the duals of isogonal polygons (see box).
A polygon is regular if and only if it is both isotoxal and isogonal. Equilateral triangles, squares and regular pentagons are convex regular polygons. More unusually, the polygon below has tenfold symmetry and is isogonal, but is not regular (note that it is not isotoxal: its sides have two different lengths).
