December 13, 2002
Squaring the circle

The TNR blog &c recently referred to squaring the circle in reference to Team Bush's new economic advisors. The point they were making was that the nominees, who have a history of advocating fiscal discipline and balanced budgets, will have a hard time reconciling those views to the tax-cut fervor of Team Bush.

All well and good, but after Matt Yglesias' admission that he didn't know what "Mayberry" was, I wondered how many folks really understand what "squaring the circle" means. So I thought I'd dust off some of my math skills and have a shot at it.

The ancient Greek mathematicians spent a lot of time doing geometry problems in which they attempted to construct various types of shapes, angles, and curves. They would do this by means of a collapsible compass (that is, a compass that would not retain its form when lifted from the paper, or in their case the ground) and an unlined straightedge (i.e., a ruler with no distances marked).

A very basic construction is a right angle to a given line. Use the straightedge to draw a line. Take the compass and put the point at one end of the line and the pencil at the other end and draw a circle, then reverse where the point and pencil are and draw another circle. Take the straightedge and draw a line from the intersection of the two circles to the original line. The angle formed by the intersection of those two lines is a right angle.

The problem of squaring a circle is actually this: Given a circle C with radius r, construct a square S whose area is equal to the area of the circle. Since the area of circle C is (pi)*(r^2), you need a square whose sides are the square root of pi times r. Since r can be defined as one unit, you need a square with side sqrt(pi).

The Greeks were never able to construct a line of such length with these methods and the additional requirement of a finite number of steps. (The Greeks understood the concept of infinity but weren't hip to the idea of an infinite sequence converging to a finite limt; if they had, then Zeno's Paradox wouldn't have been a paradox.) The problem is equivalent to the problem of whether there exists a polynomial equation with all rational coefficients and integer exponents that has the number you want (in this case, sqrt(pi), or equivalently, pi itself) as a root. All numbers that are roots of such polynomials are called algebraic. Numbers that aren't are called transcendental.

It turns out that pi is transcendental, so it cannot be constructed by Greek methods and thus the problem of squaring the circle cannot be solved. This result was not proven until 1882 by the German mathematician Lindemann, though it had been suspected for at least a century before that. In fact, in the late 1700's both the Paris Academy and the Royal Mathematical Society announced that they would no longer accept papers that purported to show a method for squaring the circle. Most of those misguided papers either disregard the Greek restrictions or assume a rational value for pi, such as 25/8.

There are two other classic Greek problems that are generally grouped with circle-squaring: doubling the dube (i.e., given a unit cube construct a cube whose volume is exactly twice that) and trisecting an angle (i.e., given an angle of d degrees, construct an angle of d/3 degrees). Both of these fail because in general cube roots and sines are not constructible. Note that being algebraic, as the cube root of two certainly is, is not sufficient to be constructible; constructible numbers are a subset of algebraic ones.

The most interesting results to come out of this was actually the work of Georg Cantor, who demonstrated that the set of transcendental numbers is not countable. A countable set is one that can be put in a one-to-one relationship with the positive integers (that is, the counting numbers - one, two, three, etc). Cantor demonstrated such a relationship for the set of algebraic numbers, then showed that the set of real numbers, which is the union of the algebraics and the transcendentals, is not countable. Since the algebraics are countable, and since Cantor also showed that a countable union of countable sets is still countable, the transcendentals must be uncountable.

In practical terms, though both sets are infinite, there's more transcendentals than algebraics. Cantor's ideas were quite revolutionary at the time, and he was thoroughly vilified by his mentor, the mathematician Kronecker, for them. The whole story is fascinating; I recommend Amir Aczel's book The Mystery of the Aleph if you want to know more.

Some other interesting links: How to square the circle by not-acceptible to-ancient-Greeks means, a proof that shows the impossibility of circle squaring by Greek methods, and this detailed history of the problem and how it was finally understood.

So that's the story of squaring the circle. Feel free to look a little smug the next time someone uses the phrase in your presence.

Posted by Charles Kuffner on December 13, 2002 to Technology, science, and math | TrackBack

Actually, the proof of the impossibility of trisecting an arbitrary angle is of a slightly different nature than that of squaring the circle of doubling the cube; it doesn't on a number (pi, the cube root of three) being unconstructable. But there's no obvious reason why you shouldn't be able to trisect, say, a 60° angle.

Instead, we use Galois theory to demonstrate it. Very, very briefly, Galois theory is concerned with the roots of polynomial equations (things like x^2 + x + 1 = 0) and what happens when you muck about with them. In particular, there's what's called "field extensions" -- if I have the field of the integers, I can "extend" it, by throwing, say, the square root of two into the mix. I believe the proof rests on demonstrating that all of the construction operations (as defined by the Greeks) extend by multiples of 2 (square roots, basically), then constructing a particular cubic (requiring an extension by a power of 3) that could be solved if you were able to trisect the 60° angle. If you could solve it, you'd've extended the field of constructables in a way that you can prove is impossible, so you can't trisect all angles.

Ugh, that's a really awful explanation. Try searching on "Galois theory" and "trisect".

Posted by: Steve on December 13, 2002 3:14 PM