Reinhardt polygon

In geometry, a Reinhardt polygon is an equilateral polygon inscribed in a Reuleaux polygon. The pairs of vertices that are farthest apart (the diameters of the polygon) include every vertex of the polygon.

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.