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Concept
Constructible polygon
Summary
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.
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 polygons with compass and straightedge? If not, which n-gons (that is, polygons with n edges) 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. 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:
A regular n-gon can be constructed with compass and straightedge if and only if n is a power of 2 or the product of a power of 2 and any number of distinct Fermat primes.
A Fermat prime is a prime number of the form
In order to reduce a geometric problem to a problem of pure number theory, the proof uses the fact that a regular n-gon is constructible if and only if the cosine is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. Equivalently, a regular n-gon is constructible if any root of the nth cyclotomic polynomial is constructible.
Restating the Gauss-Wantzel theorem:
A regular n-gon is constructible with straightedge and compass if and only if n = 2kp1p2.
Official source
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