In arts or literature, perhaps, beauty may have lost its currency in recent years as a standard of judgment or criterion for excellence, regarded as too subjective or culturally mediated. For mathematicians, however, beauty as an eternal verity has never gone out of fashion. “Beauty is the first test: there is no permanent place in this world for ugly mathematics,” wrote British number theorist Godfrey Hardy in 1941.
To get a taste of mathematical beauty, begin by heading to your favorite pub and ordering a frosty mug of beer. Place it on a paper place mat three times, forming three rings of condensation—making certain to do so in such a way that all three rings intersect at one point. Now ask your companions: How large a mug would one need to cover the other three intersection points? One nearly always assumes that only a gargantuan mug would serve that purpose. The surprise answer: the same mug! It’s a completely foolproof solution. (See figure left for two equally valid solutions; in each case, the solid circles are the first three rings; the dashed circle is the fourth ring, representing the mug covering the other three intersection points.)
This theorem was published by Roger A. Johnson in 1916. Johnson’s circle theorem demonstrates two of the essential requirements for mathematical beauty. First, it is surprising. You don’t expect the same-sized circle to show up again in the solution. Second, it is simple. The mathematical concepts involved, circles and radii, are basic ones that have stood the test of time. However, Johnson’s theorem comes up short in the beauty department in one salient respect. The best theorems are also deep, containing many layers of meaning, and revealing more as you learn more about them.
What mathematical facts live up to this high standard of beauty? German mathematician Stefan Friedl has argued in favor of Grigory Perelman’s Geometrization Theorem, for which the proof was set forth only in 2003. The theorem, which created a sensation in the world of mathematicians, advances a key step in the classification of three-dimensional topological spaces. (You can think of these spaces as possible alternate universes.) “The Geometrization Theorem,” Friedl avers, “is an object of stunning beauty.”
Boiled down to its simplest terms, it states that most universes have a natural geometric structure different from the one we learn in high school. These alternate universes are not Euclidean, or flat. The question has to do with the curvature of space itself. There are various ways of explaining what this means; the most precise one mathematically is to say that alternate universes are “hyperbolic,” or “negatively curved,” rather than flat.
Mathematicians are only beginning to grapple with the implications. Astrophysical data indicate that our own universe is flat. Yet in these alternate universes, flatness is not the natural state. According to Perelman’s theorem, our apparently flat universe constitutes a surprising exception.
Another reason that the theorem attracted international publicity has to do with the mathematician himself. In 2010, the reclusive Russian declined a million-dollar prize for his breakthrough from the Clay Mathematics Institute in Cambridge, Massachusetts. Obviously, for Perelman, mathematical beauty was not something that could be bought and paid for. Changing our understanding of the universe was reward enough.