a performance dialogue

Monday, March 17, 2008

Making a hexagon

One thought is to use the giant set of compasses to create a perfect hexagon on the floor. This is a construction that the Ancient Greeks discovered.

There is an animation on Wiki which shows how the construction is done. The point is that you can't measure anything. You can only draw a straight line or a circle with the compass.

It is possible to draw a pentagon but it is impossible to draw a 7-sided figure with this equipment. In fact if the number of sides of the shape is N then the shape can be constructed if and only if the odd primes dividing N are Fermat primes, that is primes of the form 2^2^n+1. The only Fermat primes known are 3, 5, 17, 257, 65537.

As a 19 year old, Gauss discovered a construction of the 17-sided figure. His discovery prompted him to begin a mathematical diary, one of the most important documents in the history of mathematics. Here is his construction of the 17-gon.


Dorothy said...

Interesting that you effectively have to make two circles (the arc is the beginning of a second, identical circle) to make enough co-ordinates to get the hexagon.

Dorothy said...

What about a 19-sided figure?

Marcus du Sautoy said...

you can prove that it is impossible to make a 19 sided figure. 19 is not a prime of the form 2^n+1.