[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.