I'm glad to have found this post. I discovered the Banach-Tarski theorem via Vsauce[0]. It was interesting but I couldn't get the significance of it. It either

I'm glad to have found this post. I discovered the Banach-Tarski theorem via Vsauce[0]. It was interesting but I couldn't get the significance of it. It either didn't seem like an unexpected result or too esoteric to appreciate. There's phrasing in the post that could be misunderstood (later clarified) but can leave unclarity from assumed understanding of the earlier description. > In R3, given a solid ball B of radius R, it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R. "finitely many pieces" could be confusing because though we may be talking about 6 'pieces' those pieces have an uncountable infinity of radial line slices. It's also missing the rigid motion part which is key. The part that didn't seem surprising (and is considered trivial) was in handling uncountable infinities of things. The 'number' of points on a line [0, 1) is the same as the number on a line [0, 2) so I wouldn't be surprised to map points from [0, 1) to [0, 2) filling the latter without 'gaps'. Similarly for areas. But what the theorem is saying is that this kind of mapping doesn't work in R1 nor R2 but does work in R3 (with rigid motions). The part that makes B-T surprising is that the extra volume/ball can be constructed with rigid motions of those uncountably infinite sets. This is where it seems beyond me to appreciate: that one or two balls have the same uncountably number of radial line slices is considered trivial, and mapping using rigid motions is surprising. For example, if you do the same rearrangement but instead of sets of radial lines, consider the set of points at the surface of those radial lines, we're basically working in R2. The reason why it can't be done is because if we instead of being on the surface of a sphere we're on a plane, then those same motions aren't distance preserving. Thinking geometrically doesn't seem weird: an extra dimension let's you do something you can't in lower ones. I think the algebraic description that the post makes may be clearer to appreciate the difference, and I'll be giving it another read and more thought. [0] https://www.youtube.com/watch?v=s86-Z-CbaHA https://www.youtube.com/watch?v=s86-Z-CbaHA