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If p > 2, inscribe a p-gon and a q-gon in the same circle in such a way that they share a vertex. Because p and q are coprime, there exists integers a and b such that ap + bq = 1. Then 2aπ/q + 2bπ/p = 2π/pq. From this, a pq-gon can be constructed. Thus one only has to find a compass and straightedge construction for n-gons where n is a ...
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon.Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction.
As 17 is a Fermat prime, the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796 at the age of 19. [1] This proof represented the first progress in regular polygon construction in over 2000 years. [1]
Articles related to constructible regular polygons, i.e. those amenable to compass and straightedge construction. Carl Friedrich Gauss proved that a regular polygon is constructible if its number of sides has no odd prime factors that are not Fermat primes, and no odd prime factors that are raised to a power of 2 or higher.
[2]: p. 1 They could also construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, and regular polygons of 3, 4, or 5 sides [2]: p. xi (or one with twice the number of sides of a given polygon [2]: pp. 49–50 ).
If 2 k + 1 is prime and k > 0, then k itself must be a power of 2, [1] so 2 k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023 [update] , the only known Fermat primes are F 0 = 3 , F 1 = 5 , F 2 = 17 , F 3 = 257 , and F 4 = 65537 (sequence A019434 in the OEIS ).
The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number. In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length | | can be constructed with compass and straightedge in a finite number of steps.
The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR 2 − 1 / 4 ns 2, where s is the side length and R is the circumradius. [4]: p. 73