Another question in this vein has to do with finding the right way to measure distortion, and to prove that the only way to evert a cylinder (or Mobius band, or Mobius-band-with-three-half-twists) is to introduce at least a certain amount of distortion en route. One obvious way to measure distortion of an embedded PL manifold-with-boundary is to measure the flexing happening along all the edges; the sum will be a dimensionless (angular) quantity. Or maybe we should multiply each flex-angle by the length of the edge being flexed, a la Dehn's invariant. Possibly we should draw inspiration from existing concepts in differentiable topology. Are there ways of measuring distortion of embedded or immersed manifolds, and results that (for instance) decree that if you want to evert a sphere, you need to distort it by at least a certain amount? Or maybe that's totally wrong-headed. After all, when we evert a band with some number of twists, we're not changing the local metric at all. Creases are not like cone-points. Jim Propp On Saturday, October 31, 2015, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Reconciled to knowing diddly-squat about any of these questions, I resorted to (recommended) https://en.wikipedia.org/wiki/Moebius_band Though I fail to understand properly the little it says concerning Dan's original enquiry about embedding a corrugated wide strip.
Another matter I haven't so far found discussed is the actual shape assumed by a (stiffish) flat strip when deformed into a Möbius band (isometrically, and assuming a perfect join). Presumably this can in principle be expressed as a variational problem minimising the total stress. Is there an explicit parametric solution? An implicit algebraic equation? Has there been any approximate numerical attack? Is the solution smooth, or analytic?
And totally off-topic again, I can't resist quoting a link from the Wikipedia article to "A Planetary Möbius Gear System" http://mechproto.olin.edu/final_projects/average_jo.html --- is Oskar listening?
Fred Lunnon
On 10/31/15, Dan Asimov <dasimov@earthlink.net <javascript:;>> wrote:
Would anyone care to present all the conditions in a math question all in one place at one time?
—Dan
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