Heptagons, and the politics of 19th century France.

[This is a back-issue of one of this site’s newsletters]

When I was in junior high school, my math teacher was teaching us about ruler-and-compass constructions. Do they still teach this to kids? Anyway, he’d taught us how to construct certain shapes, then informed us – the heptagon, or regular seven-sided polygon, can’t be constructed using a ruler and compass.

The next day, one of my classmates came up with this:
My teacher was baffled. He knew the heptagon couldn’t be drawn with these drawing tools, yet there it was in front of his eyes.
If you can’t see the video, here are the steps:
  • First, draw a circle, centere A, and choose a point B on the circle.
  • Draw another circle with the same radius, centered on B this time.
  • The two circles meet at two points, C and D.
  • Find the place where AB and CD meet, and call it E.
  • Then, set your compass to CE.¬†That distance will do perfectly fine as the edge length of a heptagon.
The reason this works is that the distance between C and E (relative to AB) turns out to be half the square root of three, or about 0.866025. The correct length needed to draw a heptagon is 0.867767. These numbers are so close together that you’d need incredible skill (an a very sharp pencil) to notice the difference. If your circle is 6 inches across, the length CE is wrong by less than a seventh of a millimetre, about one part in 192 of an inch.
When people say something can or can’t be constructed with a rule and compass though, they mean exactly, using a hypothetical infinitely sharp pencil and a perfectly skilled draughtsperson. In real life, you can get as close as you want to any shape you like.
It took many centuries before people knew how to prove that a heptagon was not constructible. The key insight came from a young man called √Čvariste Galois. He was an incredible mathematical genius, who sadly got involved in French politics in the early 19th century. His enemies challenged him to a duel, and he was fatally shot at the age of 20. The night before the duel was spent writing to friends, and also writing down as much mathematics as he possibly could, so that it would not be lost to the world if he lost the duel. Later mathematicians who worked through his papers found the mark of genius. An entire field of mathematics is now named after this man who died too young.
The moral of the story? Don’t get involved in early 19th century politics – mathematics is more important.

If you use a ruler and compass to construct a new point from some existing ones, there’ll be some quadratic equation that the coordinates of new point satisfies. The coefficients of the quadratic are derived from the coordinates of the points you already had.Galois’ theory shows that you can, in fact, find a polynomial with whole number coefficients for the coordinates of the new point. The degree of this polynomial will be a power of two.

So, what about the heptagon? It has a point whose coordinates are the sine and cosine of 360/7 degrees. If x is this cosine, it satisfies a cubic equation 8x3+4x2-4x-1. This doesn’t hve degree 2, or 4, or 8, or any power of two, so the heptagon is not constructible with a ruler and compass.
And there you have it!