How to Construct A Heptadecagon

Prerequisites: Complex numbers, equation of a circle

Let’s partly explain the construction of the regular 17-gon (that’s a bit less of a mouthful!) shown below:

The goal, as with any geometric construction, is to produce the desired shape using only a straightedge and a compass, a device capable of drawing a circle with any given center and radius. It’s a remarkable fact that most regular polygons cannot be constructed in this way, and nobody knew that the 17-gon could until just 200 years ago! This construction is even newer than that.

The most important idea we use in the construction is called the Carlyle circle. For any numbers s,p, the circle which has a diameter with endpoints (0,1) and (s,p) intersects the x-axis at the roots of the quadratic x²-sx+p, as long as those roots are real numbers. The Carlyle circle looks something like this:

We can see this works by observing that this circle has equation x(x-s)+(y-1)(y-p)=0.1 

So, for instance, the points C,D in our construction come from drawing the Carlyle circle centered at B=(-1/2,15/4), which has diameter from (0,1) to (-1,15/2), so their x-coordinates are the roots of  x2+x-15/2.Most of the rest of the construction proceeds by drawing Carlyle circle after Carlyle circle.

How does this help draw the regular 17-gon? Well, the particular 17-gon we draw has vertices at the seventeenth roots of unity, all the complex numbers satisfying z17=1. These have lots of excellent algebraic properties; for instance, all the solutions other than the most obvious one, z=1, sum up to -1. By using this and many other relations arising from the fact that all these roots are powers of just one of them, we are actually constructing sums of eight, then four, then just two of these seventeenth roots of unity as solutions of more and more complicated quadratics via their Carlyle circles. It’s the point T that is the sum of just two of the roots2, and from there we can get to drawing the actual vertices. 

For the details on this construction, plus a similar construction of the regular 257-gon (!) and the ideas to construct the 65537-gon (!!), see this article from the American Mathematical Monthly by Dr. Duane de Temple. You might also enjoy these Numberphile videos on a different construction, plus how Gauss came up with all this in the first place.

1For a quick check: completing the square in both x and y gives (x-s/2)2+(y-(1+p)/2)2+K=0 for some constant K, which is definitely a circle with the correct center, coming from the midpoint formula; since (0,1) and (s,p)definitely satisfy the original equation, our circle also must have the correct radius.

2Specifically, T=[(2i/17)+i(2i/17)]+[(-2i/17)+i(2i/17)]. Try to simplify this, if you know something about trigonometry and complex numbers!