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I have some questions about the video about the Banach-Tarski Paradox from the YouTube channel Vsauce:

10:09: Is this really a valid way of constructing the hyperwebster? In this order, one will never get past sequences of only "A". Shouldn't one follow an order like A, ... ,Z, AA, ... , AZ, BA, ..., BZ , ... to make sure that every sequence of letters corresponds to a well-defined position in the hyperwebster?

The same problem occurs later when he writes down the sequences of rotations, he starts with L, LL, LLL, ...

11:49: It's only shown very briefly and not explained or proven at all that using arccos(1/3) as angle of rotation makes sure that no two sequences of rotations from a given starting point end up at the same point (except poles). How can we prove this?

17:08: I find the construction of the second sphere quite confusing and don't quite see why he doesn't go with the same idea as for the first sphere. Couldn't he just rotate the "up" piece down, creating starting points, "up", "left" and "right" pieces and just combine them with the "down"-piece and then fill in the center point and poles as he explains later (or choose a countable number of points from the now left over starting points piece to fill the poles). Then he would end up two spheres AND a left over piece of starting points.

Or does he just want to use all pieces for the sake of elegance?