Recently the phrase "squaring the circle" seems to have been popping up more often than usual, perhaps in response to certain problems currently faced by the U.S. government. I could easily do a very long podcast on some of those topics, but you're probably here to talk about math, not politics. So let's get back to that phrase, "squaring the circle", which means to solve an impossible problem. It descends from a problem originating from ancient times: with no tools except a compass and a straightedge, can we construct a square with the same area as a given circle? Mathematicians struggled with this problem for thousands of years, until it was definitively proven impossible in 1882.
Before we discuss why this is impossible, let's review what the problem is. We start out with a circle of a known radius, drawn on a piece of paper. For simplicity in this discussion let's assume that the radius is 1, with the radius itself defining our unit of measurement. We are allowed to use a compass, which lets us draw a circle around any point with a given radius, and a straightedge, which allows us to draw a line connecting any two points. By the way, no cheating and using markings on the straightedge for measuring distances. Using just these tools, and without being allowed to do things like create new lines of precise lengths, we want to draw a square on the paper whose area matches that of the circle. In effect, this means that starting from a circus of radius 1, and thus an area equal to pi, we need to construct a square with area pi, and thus with each side equal to the square root of pi.
This is one of the most ancient problems known to mathematics, first appearing in some form in the Rhind Papyrus, found in ancient Egypt and dating back to 1650 B.C., though the Egyptians were happy with an approximate solution, treating pi as 256/81 rather than its real irrational value. The Greeks who followed them, starting with Anaxagoras in the 5th century B.C., had a more sophisticated understanding of mathematics, and were the first to insist on searching for a fully accurate solution rather than an approximation. Philosopher Thomas Hobbes inaccurately claimed to have squared the circle in the 1600s, leading to an embarrassing public feud between him and mathematician John Wallis. There were many other discussions and attempts to solve this problem down through the centuries, most amusingly even including two days of frantic scribbling by Abraham Lincoln at one point in the 1850s. It may have been for the best that he gave up math and returned to politics.
Nowadays, the basic concept of why circle-squaring can't be done is not that hard to grasp, with an elementary knowledge of high-school level math. Ultimately it stems from the equivalence between geometry and algebra shown by the field of analytic geometry. View the paper on which you drew the starting circle as a coordinate plane, with every point described by an x and y coordinate. Recall that both lines and circles are described by simple types of equations in such a plane. Lines are described by linear equations of the form y = mx + b, and circles are essentially polynomials in the form x^2 + y^2 = r^2. As a consequence, compass and straightedge constructions are essentially computing a combination of linear and square root functions of your starting lengths. This means that any multi-step construction is building up a set of these linear and square root operations.
But remember the target we were shooting for: we want a square of the same area as our starting circle, so each side of our square measures the square root of pi. However, pi is a transcendental number: this means you cannot get to this value by solving any set of polynomial equations with integer coefficients. Note that this is a stronger condition than being irrational: many irrationals are non-transcendental. The square root of 2, for example, is irrational, but can be derived from the simple equation "x^2 = 2." If a number is transcendental, this implies that no combination of linear and square-root operations, starting with an integer length, could ever reach a value of pi or its square root, and thus our construction of a length pi starting from a circle off radius 1 is impossible. This last part of the argument, the fact that pi is trancendental, was the hardest part of the proof, and the most recent to be put in place, proven in 1882 by German mathematician Ferdinand von Lindemann. While we can create constructions for arbitrarily precise approximations, we can never correctly derive a square with the exact same area as a given circle.
We should also point out that this impossibility is implicitly assuming we are talking about ordinary, Euclidean geometry. If our discussion is including curved spaces, as we discussed in episode 35, all bets are off. These curved spaces have many properties that upend our usual notions of geometry, such as triangles whose angles sum to more or less than 180 degrees. If instead of a flat plane we are sitting on the surface of a curved saddle or a sphere, then we can essentially get a 'free' pi by drawing a line that becomes a curve due to the space's curvature, eliminating a key component of the impossibility proof. I think we can all agree, though, that the ancient Greeks who posed the problem would consider this type of solution to be cheating.
What is most amusing about squaring the circles is that even after its impossibility was very solidly proven in 1882, enthusiastic amateurs kept on trying to "solve" the impossible problem for many years after. This has some analogues in government as well... but I'll try again to keep off that tangent. You may recall that back in episode 20, I talked about an 1897 attempt by a confused Indiana resident to legislate that pi was really equal to 3.2-- this was partially based on his supposedly successful attempts to square a circle using this value. Before you laugh too hard at him or the legislators who actually listened to him, we should note that somehow he got his bogus circle-squaring method published in the American Mathematical Monthly, which will forever be an embarrassment to that publication. In 1911, British professor E.W. Hobson wrote, "Every Scientific Society still receives from time to time communications from the circle squarer and the trisector of angles, who often make amusing attempts to disguise the real character of their essays... The statement is not infrequently accompanied with directions as to the forwarding of any prize of which the writer may be found worthy by the University or Scientific Society addressed, and usually indicates no lack of confidence that the bestowal of such a prize has been amply deserved as the fit reward for the final solution of a problem which has baffled the efforts of a great multitude of predecessors in all ages. " The circle squaring craze seems to have mostly died off in the latter part of the 20th century, but a 2003 article by NPR's "Math Guy" Keith Devlin mentioned that he still regularly received crackpot letters from circle-squarers. And even today, if you look up squaring the circle on Yahoo Answers, you'll find a post by some misguided soul who refuses to believe that the problem is unsolvable, stating that " Its possible we lack the mathematical and conceptual understanding to construct it." I guess anything is possible, but if you trust modern mathematics at all, it's pretty clear that the circle will never be squared.
And this has been your math mutation for today.