November 20, 2007
Turning a (self-intersecting) Basketball Inside out Even if you don't have time or interest for the full thing, the twenty seconds from 1:20 to 1:40 will evert your mind. By the end of the video, you'll be reminded that girls are smarter than guys, nail down some important topology concepts, and receive your contractually obligated dose of wacky mathematics humor.
If you hunger or more, read up on Smale's Paradox -- to save you the trouble, homotopy means "stretch, intersect or twist" (continuously deformable into) and regular homotopy means "however, no sharp creases" (treat the tangent bundle nicely).
That were neat.
As far as manipulating the rules, well, thats what we do, to make it all work in the world.
I don't think infinite surface area is OK. It's an immersion in a normal 3-d space, so the basketball has to sit within a finite volume (the embedding manifold is compact). You certainly wouldn't need a fractal surface for the eversion, so no infinite surface area is called for. (Now I have no idea if a Peano-type-space filling-curve-but-a-smooth-one is alllowed -- that is, if it's a 'regular homotopy' -- but it seems like it could be. Maybe the lazyweb will answer.)
As for the gaming the rules, it's a lot like figure skating and a bit like fine art. There's a lot of things you can choose to try, and strict rules imposed by nature about what you'll accomplish. Now how the scoring progresses is a communal construct, but in the special case of mathematics it's been consistently found that mathematical ideas which are elegant and interesting turn out to be scientifically useful.
Wigner wrote an essay called "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" -- it's the most interesting question in the intersection of science and philosophy that I know of.
John Doodoomeat sends his regards: man = donut.
Oh, I forgot his punchline: Kennedy was right.
What about ears and noses and eyesockets and whatnot?
He sticks salt shakers up his ass? Weird.
From today's XKCD (natch) comes this superb intersection of sex and mathematics: the taxonomy of sexual fetishes.
(Note -- it's Russell's Paradox, not Godel's illustrated there: the project is to enumerate, not to reason about formal properties of a logic system. The comic still works pedantry notwithstanding.)
phooey, maybe this belonged more in this bitrotted thread.
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I thought it was interesting but I was reminded why I could never go into mathematics. I'm sure the underlying concepts are really deep and important or something but it fills me with a mixture of arbitrariness and gaming the rules to get a result. Way back in complex analysis it always struck me as somehow wrong that you could integrate over all of complex space but cut out one point and call it a day. Same with "no cusps or pinches but it's okay to add arbitrary (up to infinite?) surface area in the form of wavelets." Never sits right with me.
posted by natedogg at 05:04PM CST on November 20