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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.
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 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).
The points of may now be used to link the geometry and algebra by defining a constructible number to be a coordinate of a constructible point. [ 8 ] Equivalent definitions are that a constructible number is the x {\displaystyle x} -coordinate of a constructible point ( x , 0 ) {\displaystyle (x,0)} [ 9 ] or the length of a constructible line ...
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]
Constructible set (topology), a finite union of locally closed sets; Constructible topology, a topology on the spectrum of a commutative ring y in which every closed set is the image of Spec(B) in Spec(A) for some algebra B over A; Constructible universe, Kurt Gödel's model L of set theory, constructed by transfinite recursion
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.
There are three regular star polygons, {16/3}, {16/5}, {16/7}, using the same vertices, but connecting every third, fifth or seventh points. There are also three compounds: {16/2} is reduced to 2{8} as two octagons , {16/4} is reduced to 4{4} as four squares and {16/6} reduces to 2{8/3} as two octagrams , and finally {16/8} is reduced to 8{2 ...