Editing Regular polygon (section)

== Constructible polygon==
{{main|Constructible polygon}}

Some regular polygons are easy to [[Compass-and-straightedge construction|construct with compass and straightedge]]; other regular polygons are not constructible at all.
The [[Greek mathematics|ancient Greek mathematicians]] knew how to construct a regular polygon with 3, 4, or 5 sides,<ref name=Bold/>{{rp|p. xi}} and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.<ref name=Bold>Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969).</ref>{{rp|pp. 49–50}} 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 [[heptadecagon|17-gon]] in 1796. Five years later, he developed the theory of [[Gaussian period]]s in his ''[[Disquisitiones Arithmeticae]]''. This theory allowed him to formulate a [[sufficient condition]] for the constructibility of regular polygons:

: A regular ''n''-gon can be constructed with compass and straightedge if ''n'' is the product of a power of 2 and any number of distinct [[Fermat prime]]s (including none).

(A Fermat prime is a [[prime number]] of the form <math>2^{\left(2^n\right)} + 1.</math>) Gauss stated without proof that this condition was also [[necessary condition|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'''.

Equivalently, a regular ''n''-gon is constructible if and only if the [[cosine]] of its common angle is a [[constructible number]]—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.