Looking for explanation of Banach-Tarski Proof, preferably by visual methods "Video, Pictures, Diagrams..."

Question

could someone please explain the four steps of Banach-Tarski? 1- Find a paradoxical decomposition of the free group in two generators. 2- Find a group of rotations in 3-d space isomorphic to the free group in two generators. 3- Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere. 4- Extend this decomposition of the sphere to a decomposition of the solid unit ball.

Answer 1

My book tells the whole story. Yes, your summary is fine. Your #1 is the easiest step: If s and t are the generators then divide the words into four sets A=W(s), B=W(sInverse), C=W(t), D=W(tInverse) where W means "words beginning on the left with". Then A union sB gives all of F_2, as does C union tD. This is a 4-piece paradox of the group.

Note that "hollow unit sphere" is incorrect. One first uses the group to get the paradox of the unit sphere (hollow) MINUS the countable collection of fixed points of all the rotations forming the group. So far, all this was done by Hausdorff. Some fairly easy set theory is used to then show that the sphere minus the countable set is equidecomposable to the whole sphere. Passage from the sphere to the ball is even easier: just use radii. That gets to the ball less the origin. But the same absorption technique used to deal with the countable set can be used to deal with (absorb) the origin.

The hardest part in all this, by far, is finding two rotations that generate the free group. For various reasons, in the revision of my book now being prepared we will use these two: {{6,2,3}, {2,3,-6},{-3,6,2}}/7 and {{2,-6,3}, {6,3,2}, {-3,2,6}}/7 . These have the advantage of acting on the rational sphere!