Friday, May 7, 2010

Our far-flung correspondents: topological observations

Again from my USMC bud.
Speaking of smoking dope, remember how I brought nice things back from Senegal and Korea? Well, this isn’t Senegal or Korea. I did, however, locate the most geometrically complex objects in Afghanistan that aren’t obviously used for smoking dope. Careful observers will note that all objects pictured have the same topology. At their present rate of development, I expect the Afghans to discover the ‘bagel’ or ‘tea cup’ topology in approximately 7000 years.
For what it's worth (not much): the objects pictured (rock, cup, saucer) are all topologically equivalent to a sphere. Imagine you have a lump of clay which you've molded into a ball. Without tearing, ripping, or poking a hole in it, you can transform that ball into any of the shapes shown. The 'bagel' or 'tea-cup' object has a hole - the hole in the bagel, the hole made by the handle of the teacup. Bagels and teacups are topologically equivalent objects.

In recent news: A Russian mathematician, Grigori Perelman, was awarded a Fields Medal for proving that a topological property of 3-dimensional spheres extends to 4-dimensional spheres (the Poincare Conjecture... with Perelman's proof, no longer a conjecture, but a theorem). (Strangely, that this property extended to 5- and higher-dimensional spheres was the easier problem.) This guy turned down the medal. On 18 March 2010, Perelman was awarded a Millennium Prize ($1M) for solving the problem. He turned down the award.

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