The longest math proof in the world has just been completed. It began in the 1970s and was worked on by 100 mathematicians. Take a look at the math equivalent of endurance running.

The Rolf Schock Award in Mathematics will go to Michael Aschbacher for helping figure out the longest proof ever made by mathematicians. In 2004, he plugged a hole in the Enormous Theorem, a proof that began in 1971 and has only recently been completed. The Enormous Theorem has taken over three decades, a hundred different people working on it, and 15,000 pages of calculations, not including the 1200 page guide that Aschbacher published to plug the hole in the proof in the first place.

What could be worth all this mathematical effort? Symmetry. Hey, if it's good enough for a supermodel's face, it's good enough for the longest proof in the world. Some shapes are symmetrical, and if you rotate them enough, they recreate the original shape. Give a square 90 degree turn, and it looks like the original square. Give it a five degree one, and it doesn't. These kinds of maneuvers on shapes can be classed into certain "families." Although the number of symmetries can go on forever, there are only a certain number of families. In 1971, it was proposed that people could find and catalog all those families.

It looks as though, at last, all of the groups have been cataloged, thanks to Aschbacher. Mathematicians have shown that the number of groups is finite, that all of them are listed, and that no more of them can ever exist. And it only takes about two trees worth of paper to prove it.

*Image: IBM Archives. Via **New Scientist**.*

## DISCUSSION

It's not that there are a finite number of shapes or even that are finite kinds of symmetry, but that each finite symmetry group can be decomposed into 'simple' groups. And now all of these simple groups have been classified. [The link says it better than I could.]

In the case of the symmetries of a regular n-sided polygon, there are 2n symmetries, n rotations and n reflections. The n reflections are also a group that can be broken down into 'cyclic groups' of prime size , where the primes divide n.

It's kind of like the simple groups are the chemical elements and they now know all the families of simple groups.