[thaumasiotes]

> The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume).

> This seems incorrect.

It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following:

1. You have a ball.

2. You cut the ball into 5 pieces in a very clever way.

3. You move the pieces around.

4. Now you have two balls, each the same size as the first.

The key, interesting part of this is in step 3, where we only use translations and rotations. Those preserve volume. (By contrast, it's easy to scale a ball of radius 2 to become a ball of radius 3, but that's not a volume-preserving transformation.) The part of the process that doesn't preserve volume is actually step 2, where we cut the ball into pieces. People find it unintuitive that this step doesn't preserve volume.

You can also cut your ball into several pieces and move the pieces around such that you end up with a much larger ball.

[joiguru]

"cut the ball into 5 pieces" is not the best description. A better one is: 2a. Split the ball into infinite pieces 2b. Divide the infinite pieces into 5 groups

[thaumasiotes]

> 2a. Split the ball into infinite pieces 2b. Divide the infinite pieces into 5 groups

Huh? The ball is already composed of infinite points. So in 2a you recognize that the ball exists, and then in 2b you cut it into pieces. But it seems superfluous to mention 2a separately.

[joiguru]

In regular natural language, cutting the ball into 5 pieces implies cutting 5 contiguous pieces.

If I say a cake is cut into 5 pieces, no person will consider that each piece contains parts from all parts of the cake.

[pfortuny]

“Cutting” it is certainly not: none of those sets is given by the zeroes of a continuous function (they would be measurable, and they cannot be).

So the paradox breaks down when you start to realize that you are not CUTTING but “choosing some points” and rearranging them. The fact that this rearrangement can be done with Euclidean moves is the surprise.

[thaumasiotes]

Come to think of it, the fact that two spheres contain the same number of points as one sphere does would seem to be closely related to why it's possible to produce two spheres from one sphere just by rearranging the points.

You can obviously produce a large sphere from a small sphere by rearranging the points, as long as you're willing to handle one point at a time -- that's what scaling is. But that requires an uncountably infinite number of translations. The Banach-Tarski theorem says we can do the same thing in only a finite number of translations.

[throwaway81523]

It is deeper than that. There is no way to do a similar duplication of a 2-dimensional disc. Why is it different in 3 dimensions? That is a property of the transformation group (rotation and translation) rather than 3-space itself.

[karmakaze]

This comment and the parent sheds some light on what makes this interesting. It didn't occur to me that we were allowing rotations and translations and that scaling is excluded. The paradox isn't about getting more from less as they're all comparably infinite, but rather being able to arrange them to be so.

[thaumasiotes]

> Why is it different in 3 dimensions? That is a property of the transformation group (rotation and translation) rather than 3-space itself.

Can you be more specific? Rotations and translations also exist in 2-space. It seems difficult to argue that this difference between 2-space and 3-space is "not a property of 3-space".

[throwaway81523]

What I mean is if you pick different transformation groups you can get different spaces where the "paradox" occurs. Likewise you can get rid of the paradox in 3-space by picking a different group.