Don't know the answer, but it brings up the question: Given a cylinder X = S^1 x [0,h] in R^3 (where S^1 is a unit circle in R^2): What is the largest h (or sup) such that X can be everted? (That is, continuously deformed through smooth surfaces such that the inside and outside change places.) Any guesses, at least? * * * The related question: What is the smallest or inf of h such that [0,1] x [0,h] can be smoothly mapped onto a Moebius band M in R^3 via f: [0,1] x [0,h] —> M with f(t,0) ~ f(1-t,h) for all t in [0,1] ? appears to have the answer inf{h} = sqrt(3). (If differentiability is assumed to be only C^1 then the answer is inf{h} = 0, though this is very hard to imagine.) —Dan
On Oct 24, 2015, at 2:47 PM, James Propp <jamespropp@gmail.com> wrote:
You can join four isosceles right triangles along hinges (I use tape) to form a square, and then join four such square panels (again with tape) to form a long rectangle, and lastly join the first and last panels together with tape to form a closed band. If you color the two sides of the band with different colors, it's a fun puzzle to turn the band inside out, exchanging the colors. I learned about this in the 80s, and amused myself by learning how to do the moves behind my back.
Does anyone know the name and/or provenance of this puzzle?=