The Banach-Tarski paradox has always appealed to me, for many of the same reasons that I've enjoyed other counter-intuitive results of math. How in the world can a solid ball be sliced into a handful of pieces which can then be reassembled into two solid balls, each as big as the original?

Well "obviously" it can't, any more than the guests in a completely-full hotel can trade rooms to make space for an arbitrary number of additional occupants—unless the solid ball, or the hotel, is an idealize infinite set. Then ordinary measures of volume fail. Send the resident of room number N to room 2*N, apply to all (infinitely many!) rooms ... and suddenly the odd-numbered rooms are vacant.

Likewise, shatter the solid ball into (infinitely spikey!) sliver-sets, choosing the super-sea-urchin fragment shapes so that rotating one of them makes it precisely fill the space that two sliver-sets formerly occupied ... and suddenly there are pieces left over that can be assembled into a second copy of the solid ball. Likewise, cut a circle into a finite number of (infinitely dusty!) pieces, slide them around, and get a perfect square of the same area.

The fun of mathematics comes with the breakdown of everyday numerical intuition, just as the fun of physics comes from the breakdown of everyday physical intuition near the speed of light, or on ultra-small scales, or in ultra-strong gravitational fields ...

(cf. RelativityPlusAstrophysics (29 Mar 2000), No Concepts At All (22 Feb 2001), LargerInside (11 May 2004), ...)

TopicScience - TopicThinking - 2005-07-25

(correlates: DyslexicMetahumor, LargerInside, NewPlace, ...)