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.

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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.