I spaced out when I originally wrote the (now-fixed) following message, having omitted mention that certain maps are to be *isometries*. Also, to avoid the Nash-Kuiper embedding theorem I've now asked that all maps be smooth (C^oo). The amended version is as follows. —Dan ---------------------------------- Given a cylinder X = S^1 x [0,h] = {{cos(theta), sin(theta), z) | 0 <= theta < 2pi, 0 <= z <= 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, smoothly deformed through smooth surfaces, each ISOMETRIC to the original, such that the inside and outside change places.) ((( Specifically: A C^oo isotopy (homotopy through embeddings) H: X x [0,1] —> R^3 such that for all p in the cylinder X (= S^1 x [0,h]) we have H(p,0) = p and H(p,1) = T(p) where for all (x, y, z) in R^3 we define T(x,y,z) := (x, y, h-z). ))) Any guesses, at least? * * * A related question: What is the smallest or inf of h such that [0,1] x [0,h] can be smoothly mapped ISOMETRICALLY 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, because of the Nash-Kuiper isometric embedding theorem. It's very hard to imagine just how this works.)