Gauss and Regular Polygons

Introduction

After studying elliptic integrals and formulas for $\pi$, we shall now focus on one of the most beautiful gems discovered by Gauss at the age of 17. Gauss proved that the construction of a regular polygon of 17 sides is possible by using an unmarked ruler and a compass only (henceforth these will be known as Euclidean tools and such constructions will be called Euclidean constructions). This is quite remarkable because since 2000 years or so from the time of Euclid the only polygons which were constructible in such a fashion were having sides 3, 4, 5, 6, 8, 10, 12, 15. Gauss added a new number in this series namely 17 and generalized his results to add further numbers. Legend has it that Gauss was so excited by this discovery that he decided to make a career as a mathematician.

The result Gauss established after deep research is the following:
A regular polygon of $ n$ sides is constructible by Euclidean tools if $ n > 2$ and
$$ n = {2}^{m}p_{1}p_{2}\ldots p_{k}$$ where $ m$ is non-negative integer and $ p_{1}, p_{2}, \ldots, p_{k}$ are distinct primes of the form
$$ p_{i} = {2}^{{2}^{j}} + 1$$ (such primes are known as Fermat primes and as of now only 5 such primes namely 3, 5, 17, 257, 65537 are known)

First of all we make some observations about the above result:

1. The conditions $ n > 2$ is obvious as we cannot have a polygon with 2 sides or less.
2. The factor $ 2^{m}$ is obvious in the formula for $ n$ because if we can construct a regular polygon of $ n$ sides then we can obviously construct a regular polygon of $ 2n$ sides. (Readers should supply a simple proof of this statement on their own.)
3. Each of the primes (other than 2, i.e. Fermat primes) in the prime factorization of $ n$ (the formula for $ n$ actually gives its prime factorization) occurs only once.
4. There may be no Fermat primes involved in the expression for $ n$ and then $ n = 2^{m}$ with $ m > 1$.
5. The condition as mentioned above is sufficient. In other words if $ n$ is of the form as specified above then the construction of polygon is possible. It does not say anything about the case when $ n$ is not in the specified form. However, it turns out that the condition is also necessary and Gauss did not have a proof of it (which is somewhat surprising because it is rather easy to establish the necessity of the condition). Thus when $ n$ is not of this particular form then a regular polygon of $ n$ sides can not be constructed by using Euclidean tools only.
6. Gauss only proves that the construction is possible using Euclidean tools, but does not provide any steps of the geometrical construction. The actual steps become quite cumbersome as $ n$ increases and the final post in this series will provide a construction of regular polygon of 17 sides.

After this brief explanation of the Gauss' result we can now ponder as to how Gauss might have proved it and follow his footsteps in this series of posts.

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