SUMS: Menzies -- The Banach-Tarski Paradox: how Group Theory crushes Geometry and proves philosophers don’t know what they’re talking about

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.