Only Five Primes Can Build a Perfect Polygon

Only Five Numbers Build a Perfect Polygon Give yourself a compass and an unmarked straightedge. You can draw a perfect triangle, square, pentagon, and hexagon with ease. But a perfect 7-sided polygon is impossible, and so is a 9-sided one. The strange part is why: which regular polygons you can build, and which you can never build, is decided entirely by a tiny list of rare prime numbers called the Fermat primes, and only five of them are known to exist. This is the story of the Gauss-Wantzel theorem. A regular polygon with n sides is constructible with compass and straightedge exactly when n is a power of two times a product of distinct Fermat primes. We walk through why the two tools can only reach lengths made from whole numbers and nested square roots, why the 7-gon needs a forbidden cube root, how 19-year-old Carl Friedrich Gauss stunned everyone in 1796 by constructing the regular 17-gon, how Euler had already broken Fermat's guess in 1732 by factoring 4,294,967,297 into 641 times 6,700,417, and how Pierre Wantzel proved in 1837 that no other polygon is possible. Every number and every polygon on screen is computed exactly: the 17-gon is drawn from true 17th roots of unity, and cos(2pi/17) really is a tower of square roots. The five known Fermat primes are 3, 5, 17, 257, and 65537. Nobody knows whether a sixth one exists. If one were ever found, a brand new buildable polygon with a colossal prime number of sides would appear along with it. CHAPTERS 0:00 The Question 1:20 What Compass and Straightedge Can Reach 2:49 The Wall at Seven 4:13 Gauss and the 17-gon 5:59 Fermat's Five Numbers 7:21 Where the Pattern Breaks 8:47 The Rule for Every Polygon 10:20 Wantzel, and Why Five Is the Whole Supply 11:59 Coda and the Open Question This is part of a series on how one person settled one hard question. Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/a... @euclideayt