On Oct 26, 2015, at 7:26 PM, James Propp <jamespropp@gmail.com> wrote:
I agree with Dan that this won't work without some additional pleats that may have to be chosen with some care. Or maybe there's no way to do it.
I'd like to ask Erik Demaine what he knows about this. Dan, may I forward your email to him?
No.
Also, can anyone provide the Gardner reference Dan has in mind?
6th Book of Mathematical Diversions, p. 63, with illustation. Gardner says there is a proof in Stephen Barr's Experiments in Topology (1964), but Barr attributes the method to Gardner and provides no proof. —Dan
Thanks,
Jim Propp
On Sunday, October 25, 2015, Dan Asimov <dasimov@earthlink.net> wrote:
Martin Gardner offered an argument (due to ?) that isn't quite valid:
Take a 100 x 1 rectangle of paper and pleat it 1001 times in the 100 direction (i.e., make 1000 folds back and forth, 100/1001 apart from each other).
You get a very thin rectangle that is 100/1001 x 1. Now bend it around with a half-twist and glue the pleated edges together to get a Moebius band. (Because there were an even number of folds, they match up.)
This may "work" with paper, but not isometrically with a genuine flat 2-manifold in R^3.
—Dan
On Oct 25, 2015, at 1:55 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Regarding Mobius bands, Dan's remark
(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.)
makes me wonder what the situation is for PL embeddings. Can a clever origamist turn ANY rectangle into a Mobius band?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun