ON A fateful night in 1832, French mathematician Evariste Galois scribbled down the details of a new kind of mathematics called group theory. The very next day he died, age 20, shot in a duel over the woman he loved. But his work lived on, becoming one of the foundations of mathematics. Now it seems Galois’ group theory may provide a surprising insight into the nature of the universe itself. And it all comes down to an unlikely suspect: the monster.
The history of the quest to find the monster is detailed in a new book by Mark Ronan (Symmetry and the Monster: One of the greatest quests of mathematics, Oxford University Press, 2006). It all begins with Galois’ group theory, the mathematical language of symmetry. An object has symmetry if it is left unchanged by some kind of transformation. For example, an equilateral triangle remains unchanged if you rotate it about its centre by 60, 120 or 360 degrees. In physics, many of the laws of nature are symmetrical. That is, they work equally well in Ohio as in Singapore and they still work if the equations undergo complex mathematical transformations.
The sets of transformations that leave the laws of physics unchanged are called symmetry groups. Physicists believe that the key to understanding the universe is to discover the symmetry groups that describe the very foundations of reality.
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Many symmetry groups have been discovered, but the ones called “finite simple groups” are most fundamental. That’s because they do for symmetry what primes do for numbers. Like the primes, finite simple groups cannot be broken down into more fundamental units, rendering them the building blocks of symmetry.
“It is so huge that its discovery took 10 years of feverish work”
Enter the monster. Many of the finite simple groups fit into a neat pattern, similar to the way atoms fit into the periodic table of elements. But 26 renegade groups refuse to fit the mould. The monster is the largest of these renegades, a group of rotations in a mind-bogglingly complex space with 196,883 dimensions. The monster is so huge that it took 10 years of feverish work before its discovery was finally announced in 1982.
Nobody knows why there are 26 of these unruly groups, but some theorists suggest they may play a role in understanding the universe at the deepest level. They believe that the best stab we have at a so-called theory of everything is an idea based on symmetry called string theory. If string theory is correct, the universe may have 26 dimensions – the same number as those mysterious groups.
Coincidence? We don’t know. All we can hope for is a genius of the stature of Galois to find the ultimate pattern and finally tame the monster.
