GaussWantzelstelling

The Gauss-Wantzel theorem, also known as the constructibility theorem, establishes the conditions under which a regular polygon can be constructed using only a compass and straightedge. Specifically, it states that a regular n-sided polygon can be constructed if and only if n is the product of a power of 2 and any number of distinct Fermat primes. A Fermat prime is a prime number of the form F_m = 2^(2^m) + 1, where m is a non-negative integer. The known Fermat primes are 3, 5, 17, 257, and 65537. Therefore, a regular n-sided polygon is constructible if n is of the form 2^k * p_1 * p_2 * ... * p_r, where k is a non-negative integer and p_1, p_2, ..., p_r are distinct Fermat primes. This theorem is a significant result in Euclidean geometry, connecting number theory with geometric constructions. It provides a definitive criterion for determining the constructibility of any regular polygon. For example, a regular 17-gon is constructible because 17 is a Fermat prime. However, a regular 7-gon is not constructible because 7 is a prime number but not a Fermat prime. The theorem was independently developed by Carl Friedrich Gauss and Pierre-Laurent Wantzel.