In geometry, aReinhardt polygon is anequilateral polygon inscribed in aReuleaux polygon. As in theregular polygons, each vertex of a Reinhardt polygon participates in at least one defining pair of thediameter of the polygon. Reinhardt polygons with${\displaystyle n}$ sides exist, often with multiple forms, whenever${\displaystyle n}$ isnot a power of two. Among all polygons with${\displaystyle n}$ sides, the Reinhardt polygons have the largest possibleperimeter for their diameter, the largest possiblewidth for their diameter, and the largest possible width for their perimeter. They are named afterKarl Reinhardt, who studied them in 1922.^[1][2]

AReuleaux 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 theReuleaux triangle. These shapes arecurves 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 theconvex 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.^[3]

If${\displaystyle n}$ is apower of two, then it is not possible to form a Reinhardt polygon with${\displaystyle n}$ sides. If${\displaystyle n}$ is anodd number, then theregular polygon with${\displaystyle n}$ sides is a Reinhardt polygon. Any othernatural number must have an odddivisor${\displaystyle d}$, and a Reinhardt polygon with${\displaystyle n}$ sides may be formed by subdividing each arc of a regular${\displaystyle d}$-sided Reuleaux polygon into${\displaystyle n/d}$ smaller arcs. Therefore, the possible numbers of sides of Reinhardt polygons are thepolite numbers, numbers that are not powers of two. When${\displaystyle n}$ is an oddprime number, or two times a prime number, there is only one shape of${\displaystyle n}$-sided Reinhardt polygon, but all other values of${\displaystyle n}$ have Reinhardt polygons with multiple shapes.^[1]

The diameter pairs of a Reinhardt polygon form manyisosceles triangles with the sides of the triangle, with apex angle${\displaystyle \pi /n}$, 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${\displaystyle n}$. The diameter of the polygon (the longest distance between any two of its points) equals the side length of these isosceles triangles,${\displaystyle 1/2\sin (\pi /2n)}$. Thewidth of the polygon (the shortest distance between any two parallelsupporting lines) equals the height of this triangle,${\displaystyle 1/2\tan (\pi /2n)}$. These polygons are optimal in three ways:

• They have the largest possible perimeter among all${\displaystyle n}$-sided polygons with their diameter, and the smallest possible diameter among all${\displaystyle n}$-sided polygons with their perimeter.^[1]
• They have the largest possible width among all${\displaystyle n}$-sided polygons with their diameter, and the smallest possible diameter among all${\displaystyle n}$-sided polygons with their width.^[1]
• They have the largest possible width among all${\displaystyle n}$-sided polygons with their perimeter, and the smallest possible perimeter among all${\displaystyle n}$-sided polygons with their width.^[1]

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).^[7]

The${\displaystyle n}$-sided Reinhardt polygons formed from${\displaystyle d}$-sided regular Reuleaux polygons are symmetric: they can be rotated by an angle of${\displaystyle 2\pi /d}$ to obtain the same polygon. The Reinhardt polygons that have this sort of rotational symmetry are calledperiodic, and Reinhardt polygons without rotational symmetry are calledsporadic. If${\displaystyle n}$ is asemiprime (the product of twoprime numbers), or the product of apower of two with an oddprime power, then all${\displaystyle n}$-sided Reinhardt polygons are periodic. In the remaining cases, when${\displaystyle n}$ has at least two distinct odd prime factors and is not semiprime, sporadic Reinhardt polygons also exist.^[2]

For each${\displaystyle n}$, there are only finitely many distinct${\displaystyle n}$-sided Reinhardt polygons.^[3] If${\displaystyle p}$ is the smallest prime factor of${\displaystyle n}$, then the number of distinct${\displaystyle n}$-sided periodic Reinhardt polygons is$${\displaystyle \frac{p{2}^{n/p}}{4n}(1+o(1)),}$$where the${\displaystyle o(1)}$ term useslittle O notation. However, the number of sporadic Reinhardt polygons is less well-understood, and for most values of${\displaystyle n}$ the total number of Reinhardt polygons is dominated by the sporadic ones.^[2]

The numbers of these polygons for small values of${\displaystyle n}$ (counting two polygons as the same when they can be rotated or flipped to form each other) are:^[1]

${\displaystyle n}$:	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
#:	1	0	1	1	1	0	2	1	1	2	1	1	5	0	1	5	1	2	10	1	1	12

• Biggest little polygon, the polygons maximizing area for their diameter

• Types of polygons

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