Quandaries
and Queries 



A construction of a pentagon is in the answer to answer to a previous question For the second construction, it's a "heptagon." We generally use Greek words for geometric objects. The heptagon is the smallest regular polygon that cannot be constructed with ruler and compass. Gauss's big discovery (when he was a 19year old kid who hadn't yet decided whether he would become a mathematician when he grew up) was that a regular ngon can be constructed if and only if n = 2^{k}*p*q*... where p, q, ... are DISTINCT Fermat primes  prime numbers of the form 2^{m} + 1, and k is a nonnegative integer. That is, n must be a power of 2 times a product of distinct Fermat primes. The number of Fermat primes can be 0, 1, 2, 3,... Thus there is a construction of a pentagon since 5 = 2^{0}*5 and 5 is a Fermat prime, but no construction of a heptagon. At this young age Gauss also gave a construction of a 17gon. Chris 

