Constructible polygon
Some regular polygons are easy to construct with compass and straightedge; others are not.
49–50 This led to the question being posed: is it possible to construct all regular polygons with compass and straightedge?
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.
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 the product of a power of 2 and any number of distinct (unequal) Fermat primes.
The number of Fermat primes involved can be 0, in which case n is a power of 2.
is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.
As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the modulo-2 Pascal's triangle, minus the top row, which corresponds to a monogon.
This pattern breaks down after this, as the next Fermat number is composite (4294967297 = 641 × 6700417), so the following rows do not correspond to constructible polygons.
In general, if there are q Fermat primes, then there are 2q−1 odd-sided regular constructible polygons.
In the light of later work on Galois theory, the principles of these proofs have been clarified.
This number lies in the n-th cyclotomic field — and in fact in its real subfield, which is a totally real field and a rational vector space of dimension where φ(n) is Euler's totient function.
Wantzel's result comes down to a calculation showing that φ(n) is a power of 2 precisely in the cases specified.
In this way the result of Gauss can be understood in current terms; for actual calculation of the equations to be solved, the periods can be squared and compared with the 'lower' periods, in a quite feasible algorithm.
If n = pq with p = 2 or p and q coprime, an n-gon can be constructed from a p-gon and a q-gon.
Thus one only has to find a compass and straightedge construction for n-gons where n is a Fermat prime.
A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if
