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# The constructibility of regular polygons

You know about polygons. Shapes if you prefer one syllable words. What you might not know but should is that not regular polygons can be constructed with just a straightedge and compass. Here are some of the ones that can.

(Source)

If you watch the video with no other knowledge of the topic, some of them might be surprising. Why is a regular pentagon constructible but not a heptagon? How on earth can a 17-gon be constructible?

As always, the answer is math. It turns out that a regular n-gon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat primes.

Wait, Fermat prime? Let’s take a look. A Fermat number (named after Pierre de Fermat of Fermat’s Last Theorem and Fermat’s Little Theorem) is a number of the form 2^(2^n)+1, where n is a nonnegative number. If 2^(2^n)+1 is prime, then the number is called a Fermat prime. Only the first five Fermat numbers (n=0,…,4) are known to be prime. Checking those is an exercise for you, but have fun checking the n=4 case. No one knows if those are the only Fermat primes. If you solve that problem, you could be famous. Get to work. This is a message for all of us, myself included.

But you’ve probably spotted something. What about squares? Octagons? 16-gons? Any n-gons where n is a power of two? All the n-gons that are powers of two can be constructed, and that’s because you don’t have to multiply by a Fermat prime to accomplish that.

Now have fun constructing those polygons.

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