Type: Seminar

Modified: Wed 6 Apr 2011 1224

Distribution: World

Expiry: 7 Apr 2011

Auth: duncans@60-240-60-196.tpgi.com.au (dsut7890) in SMS-WASM

The ever entertaining Max Menzies has returned from Cambridge to give a talk. Please note the unusual location, Carslaw 175. Note the change from the previously advised location. Abstract: The Banach-Tarski Paradox is a theorem that says something utterly bizarre, even stranger than Yinan. Roughly speaking, it is possible to disassemble a tennis ball (a solid unit sphere in 3-dimensional space) into a finite number of parts, move these parts around by rotations and translations, and put them back together to form TWO solid unit spheres. Dead serious. In my talk, I will first try to explain how on earth this can be, for this appears to contradict any intuitive notion of a "law of conservation of mass/volume." I will discuss the non-geometric nature of the reals and the fact that an intuitive notion of volume for all subsets of 3D space simply doesn’t exist. Just to annoy philosophers, I will explain how we should expect such notions to fail! I will then (almost) prove this theorem, and explain exactly why it is true: it all comes down to the (nasty) group of rotations of 3D space. If I have time, I will remark something even more surprising: the Banach-Tarski paradox does not apply to 2-dimensional or 1-dimensional space. For these, it is possible to get some sort of notion of volume. Again, this is all due to the group of rotations of 2D space. Omnia vincit group theorema.