Quandaries and Queries


I don't exactly know the level probably Gr. 12 to university
Gr. 10 Student
Our teacher just finished the constructions unit, and he mentioned briefly about odd sided figures such as pentagons and septagons, only that they're very hard.  My question is, how do you draw, with a compass and a straight edge, a pentagon and septagon? 



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 19-year old kid who hadn't yet decided whether he would become a mathematician when he grew up) was that a regular n-gon can be constructed if and only if n = 2k*p*q*... where p, q, ... are DISTINCT Fermat primes -- prime numbers of the form 2m + 1, and k is a non-negative 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 = 20*5 and 5 is a Fermat prime, but no construction of a heptagon.

At this young age Gauss also gave a construction of a 17-gon.



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