Credit: Mark Weiss
Fearless Symmetry
By Avner Ash and Robert Gross
(Princeton University Press)
Symmetry and the Monster
By Mark Ronan
(Oxford University Press)
Compared to other key concepts of contemporary mathematics and physics—infinity, uncertainty, undecidability, relativity—the notion of symmetry might seem a bit pedestrian. Things look the same as their reflection in the mirror—big whoop!
But symmetry conditions our understanding of the universe more completely than any of these other ideas. It would not be far off to say that our basic understanding of what the universe is depends, fundamentally, on the symmetries we believe it possesses. In the Newtonian universe, the symmetries were pretty simple—essentially, physics didn't change if you stood on a moving boat and everything worked perfectly. Except that it didn't actually describe the universe we live in. All the famous heart-worrying features of relativity—the contraction of objects moving at high speed, the constancy of the speed of light—are consequences of the fact that the universe obeys a different set of symmetries than the ones the Newtonian physicists imagined. These laws involve more complicated transformations, called the Lorentz symmetries, in which space and time can't be separated.
But here's where it starts getting tricky. We don't know—even now, with all the increased orders-of-magnitude in our powers of observation—what the symmetries of the universe truly are. We are still trying to understand what they could be. When string theorists contemplate their proposed 11-dimensional universe, they aren't (yet) asking whether the theory matches the real world—they're asking whether there even exists a theory to be tested. Which is where mathematicians come in. We get the fun job of working out what kinds of symmetries are mathematically possible, without the pressure of having to show that our constructions actually conform to anything in the universe.
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Two new books, Mark Ronan's Symmetry and the Monster and Avner Ash and Robert Gross's Fearless Symmetry, treat different aspects of this attempt to discover and classify all possible symmetries. Ronan's topic is the classification of the finite simple groups and the development of the so-called "Monster group." Ash and Gross take on the whole grand subject of Galois representations and reciprocity laws. If the words in the preceding two sentences mean nothing to you, then you've begun to understand the terrific challenges these authors are taking on. Mathematics has proceeded so far and so quickly that a survey of contemporary study means catching up on a century or two of homework before being able to understand what someone proved last Tuesday.
Image Credit: Marian Bantjes
That said, it's not so hard to describe what mathematicians mean by symmetry. A symmetry is no more or less than a way of transforming a mathematical object which is reversible. For instance, if our object is the set {0,1}, a perfectly good symmetry is the transformation T which exchanges 0 and 1. It's easy to reverse this transformation—just switch the two numbers back to their original positions! Similarly, there is a symmetry of the sphere (i.e. the surface of the Earth) which transforms each point into its antipode, the point directly opposite on the globe. Call this symmetry A. The fundamental insight that animates both these books is that it is the symmetries themselves, not the objects on which the symmetries operate, that are the true entities of interest. The antipodal symmetry is the antipodal symmetry whether it is transforming the surface of the Earth, the Moon or a tennis ball. What's more, the symmetry A and the symmetry T are themselves quite similar: when you execute them twice, the overall effect is the same as doing nothing at all. So though the surface of the Earth and the set {0,1} look nothing alike, they share this very profound property.