Since the Greeks, it was known how to construct regular polygons with 3, 4, 5 and 15 sides with a ruler and compass, and all that come out of them doubling the number of sides (6, 8, 10, 12…). For 2000 years, no one found another.

Gauss showed that constructing the $n$ sided polygon is the same as calculating $\cos\frac{2\pi}{n}$ using only integers, the four operations and square roots. And ruler and compass know how to take root square: it is exactly the construction that we saw in the chapter on Descartes. For 17 sides, he reached

$$\begin{aligned} \cos\frac{2\pi}{17} = {} & \frac{-1 + \sqrt{17} + \sqrt{34 - 2\sqrt{17}}}{16} \\ & + \frac{1}{8}\sqrt{17 + 3\sqrt{17} - \sqrt{34 - 2\sqrt{17}} - 2\sqrt{34 + 2\sqrt{17}}} \\ \approx {} & 0{,}93247. \end{aligned}$$

It only has square roots, one inside the other. Therefore, the polygon with 17 sides can be constructed.

What is the rule? Gauss discovered that the $n$ sided polygon can be constructed when $n$ is a power of 2 times different Fermat primes: the primes of the form $2^{2^k} + 1$, which are 3, 5, 17, 257 and 65 537. These are the same Fermat numbers from the chapter Euler, which showed that the next one, $2^{32}+1$, is not a prime. In 1837, Pierre Wantzel proved that no other polygons can be constructed.

Choose the number of sides. Green: you can build it with a ruler and compass. Red: no way.

Up to 30 sides, the constructibles are: 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24 and 30. The one on 7 sides, for example, it cannot be built. See also the post about regular polygons.

All topics by Carl Friedrich Gauss · ← The sum of 1 to 100 · Clock arithmetic →

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