Polygon

A polygon is any polytope of rank two. These are usually realized in two dimensions, as shapes bounded by straight lines. Polygons can be convex or nonconvex.

Polygons that aren't compounds consist of a single circuit of vertices and edges. They always have the same amount of vertices as edges. As such, polygons may be characterized by any of these numbers. A polygon with n vertices or edges is called an n-gon, where n is replaced by the appropriate Greek root. The single exception to this is the triangle, although the name "trigon" is also valid though uncommon.

The simplest possible non-degenerate polygon is a triangle. The digon is a valid abstract polytope, although it is usually disallowed geometrically as it can only be realized with curved or coinciding edges. Monogons aren't valid abstract polytopes, although they may make sense under more general notions of polytopes.

Abstractly, all polygons are regular. Geometrically, the regular polygons are those with congruent edges and equal interior angles. There's infinitely many geometrically regular (non-compound) polygons, one for each number of sides starting from 3. This is unlike the higher-dimensional geometrically regular polytopes, of which there are only finitely many for each dimension.

All polygons may be shown to be abstractly self-dual. However, there are names used to distinguish geometrically dual pairs of polygons, such as an isosceles trapezoid and a kite, which are duals to each other.

The possible symmetries of a polygon include no symmetry (scalene triangle), central inversion symmetry (parallelogram), mirror symmetry (isosceles triangle), and dihedral symmetry (square).

Polygonal symmetries can exist in higher dimensions, such as pyramidal symmetries, duoprismatic symmetries, and step prism symmetries. There also exist Petrie polygons, which are skew polygons existing in three dimensions or higher. For example, the cube has a regular skew hexagon as its Petrie polygon.

In our three-dimensional world, polygons can occur naturally, such as hexagons in snowflakes, or artificially, such as in art and in coins.