On a more mathematically substantive topic, can the proof of Borsuk-Ulam on S^2 that Vsauce presents be made rigorous without a lot of extra machinery? Neither of the proofs at https://en.m.wikipedia.org/wiki/Borsuk–Ulam_theorem is as simple as one might hope. Jim Propp On Tuesday, October 4, 2016, James Propp <jamespropp@gmail.com> wrote:
Michael at Vsauce, in his superb video on fixed points ( https://m.youtube.com/watch?v=csInNn6pfT4), asserts that the reason you eventually end up with 4 if you iterate the operation "spell the word and count the letters" is that 4 is the only fixed point of this map. This neglects the possibility of cycles. Can anyone find a language in which the spell-and-count map contains one or more cycles of length greater than 1?
There is some ill-definedness of the spell-and-count map in English, and probably other languages too; e.g., 101 is both "one hundred one" and "one hundred and one". There's also the matter of the British billion vs. the American billion. All such variants are legitimate for purposes of my question.
Jim Propp