Compass-and-Straightedge Construction of a Regular Heptadecagon

Why can a regular 17-gon be constructed with a compass and straightedge? How is it done? Don’t be in a hurry; please look at the explanation below:

A necessary and sufficient condition for a regular prime polygon to be constructible with a compass and straightedge is that the number of sides of the polygon must be a Fermat prime. In other words, only regular polygons with 3, 5, 17, 257, and 65537 sides can be constructed using a compass and straightedge; other regular prime polygons cannot (unless we discover another Fermat prime).

The compass-and-straightedge construction of the regular 17-gon was derived by Gauss in 1796, which led to his determination to become a mathematician. Regarding Fermat primes, they refer to prime numbers of the form 2^{2^n}+1. Initially, Fermat believed that for all n, numbers of this form were prime. However, as if by a joke of nature, currently only n=0, 1, 2, 3, 4 have been found to yield prime numbers (2^{2^n}+1), while the rest are composite.

Richelot provided the compass-and-straightedge construction for the regular 257-gon, which filled 80 full pages. Hermes provided the construction for the regular 65537-gon; this manuscript filled an entire suitcase and is now kept at the University of Göttingen in Germany. This is the most tedious compass-and-straightedge construction in history.

To prove that a regular 17-gon is constructible is actually quite simple, because we have: \resizebox{0.95\linewidth}{!}{$ \displaystyle \cos\frac{2\pi}{17}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}}{16} $}

The necessary and sufficient condition for compass-and-straightedge construction is: A length can be constructed if it can be derived from the unit length 1 through a finite number of additions, subtractions, multiplications, divisions, and square root extractions. Since \cos\frac{2\pi}{17} satisfies this condition, the regular 17-gon can be constructed with a compass and straightedge. (As for \cos\frac{2\pi}{17}, it is derived through a series of trigonometric calculations; readers might want to try it themselves, as I have not yet found the detailed process.)

Having said all that, let’s get to the main topic. Here are the methods for the compass-and-straightedge construction:

1. GIF Version (This seems a bit too complex)

2. Flash (1)

3. Flash (2)

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