Construction of Regular Polygons
The problem of which regular polygons can be constructed by ruler and compass alone goes back to the ancient Greeks. Some regular polygons (e.g. a regular pentagon) can be constructed with a straightedge (without any markings) and compass, others cannot. Carl Friedrich Gauss made the first new progress on the problem when he constructed the regular 17-gon in 1796. He later showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes.
In order to explain Gauss's Theorem, we need to understand about Fermat numbers, which are defined as numbers of the form 22n+1,
where n is natural number or zero. The first 5 Fermat numbers are: 3, 5, 17, 257, 65537.
Gauss' Theorem on the constructibility of regular polygons says that a regular n-gon is constructible by straightedge and compass alone if and only if n = 2kp1...pt , where k and t are nonnegative integers and pi are distinct prime Fermat numbers.
By this theorem, we can construct all of the first 8 regular polygons from an equilateral triangle up to a decagon, with the exception of a heptagon. As 7 is not a Fermat prime nor has it any factors which are Fermat primes, a regular heptagon therefore cannot be constructed by straightedge and compass alone.
The construction of the regular 17-sided polygon is an inscription on Gauss' tomb.
- This result is recorded in Section VII of Gauss's Disquisitiones Arithemeticae published in 1801. Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was proved by Pierre Wantzel in 1837
- Note that all of these are prime and Fermat conjectured in 1640 that all the Fermat numbers are prime. Rather surprisingly it wasn't until 1732 that Euler pointed out that the next Fermat number 4294967297 is not prime. It is divisible by 641. In fact the first 5 are the only know prime Fermat numbers and it seems reasonable that there are no others