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In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known.
The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65,537 is a Fermat prime , being of the form 2 2 n + 1 (in this case n = 4).
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]
[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 ).
The regular 257-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 257 is a Fermat prime , being of the form 2 2 n + 1 (in this case n = 3).
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.
However, it is constructible using neusis, or an angle trisector. The following is an animation from a neusis construction of a regular tridecagon with radius of circumcircle O A ¯ = 12 , {\displaystyle {\overline {OA}}=12,} according to Andrew M. Gleason , [ 1 ] based on the angle trisection by means of the Tomahawk (light blue).
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.