Reinhardt polygon

sides exist, often with multiple forms, whenever

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.

[1][2] A Reuleaux polygon is a convex shape with circular-arc sides, each centered on a vertex of the shape and all having the same radius; an example is the Reuleaux triangle.

These shapes are curves of constant width.

Some Reuleaux polygons have side lengths that are irrational multiples of each other, but if a Reuleaux polygon has sides that can be partitioned into a system of arcs of equal length, then the polygon formed as the convex hull of the endpoints of these arcs is defined as a Reinhardt polygon.

Necessarily, the vertices of the underlying Reuleaux polygon are also endpoints of arcs and vertices of the Reinhardt polygon, but the Reinhardt polygon may also have additional vertices, interior to the sides of the Reuleaux polygon.

is a power of two, then it is not possible to form a Reinhardt polygon with

is an odd number, then the regular polygon with

sides is a Reinhardt polygon.

Any other natural number must have an odd divisor

sides may be formed by subdividing each arc of a regular

is an odd prime number, or two times a prime number, there is only one shape of

have Reinhardt polygons with multiple shapes.

[1] The diameter pairs of a Reinhardt polygon form many isosceles triangles with the sides of the triangle, with apex angle

, from which the dimensions of the polygon may be calculated.

If the side length of a Reinhardt polygon is 1, then its perimeter is just

The diameter of the polygon (the longest distance between any two of its points) equals the side length of these isosceles triangles,

The curves of constant width of the polygon (the shortest distance between any two parallel supporting lines) equals the height of this triangle,

These polygons are optimal in three ways: The relation between perimeter and diameter for these polygons was proven by Reinhardt,[4] and rediscovered independently multiple times.

[5][6] The relation between diameter and width was proven by Bezdek and Fodor in 2000; their work also investigates the optimal polygons for this problem when the number of sides is a power of two (for which Reinhardt polygons do not exist).

-sided Reinhardt polygons formed from

-sided regular Reuleaux polygons are symmetric: they can be rotated by an angle of

The Reinhardt polygons that have this sort of rotational symmetry are called periodic, and Reinhardt polygons without rotational symmetry are called sporadic.

has two distinct odd prime factors and is not the product of these two factors, sporadic Reinhardt polygons also exist.

is the smallest prime factor of

However, the number of sporadic Reinhardt polygons is less well-understood, and for most values of

the total number of Reinhardt polygons is dominated by the sporadic ones.

[2] The numbers of these polygons for small values of

(counting two polygons as the same when they can be rotated or flipped to form each other) are:[1]