Reinhardt polygon

In geometry, a Reinhardt polygon is a convex polygon in which the triangles formed from each edge and the farthest point from the edge are all congruent and isosceles. The sides of these isosceles triangles form the farthest pairs of its vertices (its diameters) and include every vertex of the polygon. Reinhardt polygons may be constructed from certain Reuleaux polygons, curves of constant width made up of circular arcs of constant radius, by subdividing and straightening the arcs of a Reuleaux polygon into equal-length line segments.

The number of sides of a Reinhardt polygon can be any positive integer that is not a power of two. For any odd number ${\displaystyle n}$, the regular ${\displaystyle n}$-gon is a Reinhardt polygon. There is only one shape of Reinhardt ${\displaystyle n}$-gon when ${\displaystyle n}$ is either a prime number or twice a prime number, but for other values there are multiple different Reinhardt ${\displaystyle n}$-gons. A formula counts the Reinhardt ${\displaystyle n}$-gons with rotational symmetry, but many Reinhardt polygons are asymmetric.

Among all polygons with ${\displaystyle n}$ sides, the Reinhardt polygons have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter. They are named after Karl Reinhardt, who studied them in 1922.