TITLE: Banach Tarski -- any demonstration?
QUESTION [8 upvotes]: Is there any where to watch a video of a ball being decomposed into 5 pieces that are then translated and rotated to create two balls? 
How is this even possible without stretching? 
Is it possible to give a simple proof (along the lines of why it is impossible to cube the cube because the smallest cube will have no space above it to put a cube any larger than itself) 
In other words -- how would I convince a 6 year old of this? Further questions : 1) If the sets are not measurable, why do there even need to be 5 of them, why not just 2?

REPLY [6 votes]: That demo of mine cited by amWhy shows that a paradox in the whole hyperbolic plane is possible using pieces that are simple (hyperbolic) triangles. These are indeed measurable sets. That is ok, since the total measure of H^2 is infinite, so it just shows that infinity = 2 infinity. More precisely, it shows that a certain type of measure  in H^2 (a type of total measure 1, which DOES exist in R^2) cannot exist. Additionally, it is a nice pictorial representation of the paradox in the abstract group.
The material is discussed in detail in my book The Banach-Tarski Paradox (almost 30 years in print: I am now starting work on a second edition with G. Tomkowicz!) and also in Chapter 19 of my Mathematica in Action (3rd ed.)
And I admit I was very very terse in that demo on the WRI site (from several years ago)  not even explaining, as I did above, what the consequences of such a constructive paradox in the hyperbolic plane are. More on the demo: It is really showing the Hausdorff paradox: Each of the three colored sets, orange, blue, and green, are one third of the space, since they are all congruent. But moving our viewpoint a little, one sees that two of the sets are congruent to the third, so each set is also a half of the plane. So 1/2 = 1/3 (if a congruent measure of total measure 1 existed; so such a measure does not exist).