The Paradox That Says 1 Sphere = 2 Spheres

追加: 2026-06-07

[00:00:00]The Banach-Tarski paradox says that a solid sphere can be split into a finite number of pieces and without any stretching or adding material, we can reassemble it into two identical copies of itself. Now, this sounds impossible and the secret lies in a particular type of structure we call a free group. Based on two generators, and its elements [music] are all finite words that we can build from these two symbols and their inverses.

[00:00:24]We can represent all of these words on an infinite tree that we call the Cayley graph.

[00:00:30]The words form branches which go on forever.

[00:00:34]Now, if we multiply the branch beginning with A inverse by A, we reproduce almost the entire tree itself. We reproduce all those words that do not begin with A.

[00:00:45]And so, using only two of the branches, >> [music] >> we have reproduced almost the entire tree itself. And we can do a similar thing with the other two branches.

[00:00:54]Now, it turns out that this particular structure, the free group structure, >> [music] >> is hiding inside the set of rotations on the sphere.

[00:01:02]With two carefully chosen rotations, we can recreate the free group structure, which is what lies behind Banach-Tarski.

[00:01:09]Now, it's more complicated than this.

[00:01:11]So, if you want a full explanation, please check out full video on my channel.