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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 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).
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]
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).
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.
Dih 15 has 3 dihedral subgroups: Dih 5, Dih 3, and Dih 1. And four more cyclic symmetries: Z 15, Z 5, Z 3, and Z 1, with Z n representing π/n radian rotational symmetry. On the pentadecagon, there are 8 distinct symmetries. John Conway labels these symmetries with a letter and order of the symmetry follows the letter. [3]
Regular triacontagon with given circumcircle. D is the midpoint of AM, DC = DF, and CF, which is the side length of the regular pentagon, is E 25 E 1.Since 1/30 = 1/5 - 1/6, the difference between the arcs subtended by the sides of a regular pentagon and hexagon (E 25 E 1 and E 25 A) is that of the regular triacontagon, AE 1.
Constructible polygons (19 P) P. Polyforms (22 P) Polygons by the number of sides (2 C, 26 P) S. Star polygons (41 P) Pages in category "Types of polygons"