Constructible polygon

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.

Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not?

Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:

(A Fermat prime is a prime number of the form ${\displaystyle 2^{(2^{n})}+1.}$) Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem.

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