[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.