Polygon

A polygon is a closed shape, usually in two dimensions, bounded by straight lines. It is a rank two polytope. It can be convex or nonconvex. Regular polygons are a special subset of polygons with congruent edges and interior angles, and are infinite in number, unlike their higher-dimensional equivalents. The simplest possible non-degenerate polygon is a triangle.

All polygons are abstractly self-dual, because the number of vertices are always the same as the number of edges. 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, which make use of such symmetries in their basic topology. 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.