Take a rubber band and loop it round an orange. Slide it, shrink it, and it will always draw down to a point without leaving the surface. Now loop it round the handle of your mug. However you tug, it is caught — the hole is in the way. That difference is how a mathematician tells a sphere from a doughnut without ever stepping back to look at the whole thing. Poincare asked, in 1904, whether the same test worked one dimension up: if every loop on some three-dimensional shape can be pulled to a point, must it be a sphere? The kettle-table version is easy. The real one took a century.

On this day in 2006, Grigori Perelman was awarded the Fields Medal for proving it, and declined. He had put the work up as three preprints, free to read, building on Richard Hamilton's method of letting a shape smooth itself out like heat spreading through metal. No journal, no fanfare, no fee. He had already stepped away from his post, and has kept his privacy since.

The medal below carries a profile of Archimedes and a Latin motto about rising above oneself to grasp the world. I have known a few people happiest at the bench and least happy at the podium, and I was mildly one of them. Whatever his reasons, the proof was given away to anyone who wanted it, and it is still there.