That helps; thanks, Andy! (Now I understand what Fred was saying earlier in this thread.) Is there a way to see this even more directly? I think the claim in question may be strictly weaker than the nontriviality of the fundamental group of O(3). It smells like a two-dimensional fact to me. Jim Propp On Tuesday, October 27, 2015, Andy Latto <andy.latto@pobox.com> wrote:
Place a frame (basis for R3) at every point of the rubber band. Choosing a favorite frame, frames correspond to elements of O(3). So a position of the band gives a map from the circle to O(3). Place the frames initially all the same way, so that this map is initially the constant map to the identity. As you move the rubber band through space, you also move the map from the circle to O(3). But the fundamental group of O(3) has two elements, so you there are two types of closed paths in R3, and you can't deform one type into the other. Positions with an odd number of loops map to homotopically trivial loops in O(3); positions with an even number of loops to non-homotopically trivial loops.
Andy
On Tue, Oct 27, 2015 at 10:57 AM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Question 1: Is there any mathematical/physical theory governing stable shapes formed by rubber bands and things like rubber bands, as shown in http://mathenchant.org/Rubber-band.jpg ?
Note that the rubber band wraps five times around the vertical axis (imagine sticking a pencil through the shape), in keeping with the whole odd-number-of-loops phenomenon.
James Tanton, in puzzle 12.2 of his book "Solve This" (see pages 28 and 144), explains the parity phenomenon as follows: "There is only one way to produce loops in a band of paper that is sitting on a table top while maintaining its vertical 'walls'. Pinch a fold of paper into the center of the loop. Pick up the end of this fold and flip it back over to the outer edge of the band to form two loops (with upright walls) within the band. Any loop you produce in the band must be balanced by another loop counteracting the effects of the production of the first. For this reason, two loops appear in the procedure illustrated; and, in arbitrary manipulations, only even numbers of loops can appear. Adjusting the paper (or the rubber band) so as to "stack" these loops along the original circuit of paper thus produces an *odd* number of loops to wrap around a pencil."
Question 2: How can one make this proof rigorous? Is there a way to make it simultaneously intuitive and rigorous?
To see the passage in question, Google ""rubber band" pencil tanton". I believe Tanton gives some references related to the odd parity phenomenon, but I am not sufficiently adroit with Google to gain access to the relevant pages, and I don't have a copy of Tanton's book; can any of you provide leads while I'm waiting for my copy to arrive at the library?
Thanks,
Jim Propp
On Tue, Aug 25, 2015 at 4:12 PM, Fred Lunnon <fred.lunnon@gmail.com <javascript:;>> wrote:
All tied in with the soup-plate trick, that wretched puzzle involving untangling the lines joining a ball to the corners of a surrounding cube, electron spin, Spin groups, and the simply-connected double cover of the 3-D Euclidean isometry group.
None of is of the slightest use when wrestling with your band-saw blade, bicycle tyre or whatever; and I may not be of much assistance with said tent, the design of which is unfamiliar to me.
The general approach with a stiffish band is (1) grasp horizontal loop at two diametrically opposite points A,B ; (2) rotate B through pi about axis AB , causing collision at C ; (3) simultaneously, allow B to rise and fold over, forming one-third size sub-loop; (4) allow A to rotate naturally through pi , causing collision at D ; (5) simultaneously, allow A to rise and fold over, forming one-third size sub-loop; (6) relax --- you (should) now have 3 unstressed sub-loops.
WFL
On 8/25/15, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
I took a look at https://www.youtube.com/watch?v=f-7x8X6I3RA because I needed to fold a kid's camping tent --- a task that poses similar differential-geometric challenges (but without the sharp teeth).
The solution Paul Sellers advocates uses a table or other external object. Is there an easy-to-describe hands-only two-person solution?
Also: Is there a write-up anywhere of the mathematics behind folding a band? I'm guessing that it's the kind of situation where you can get a (2n+1)-to-1 fold for any n. But I don't even see how the 3-to-1 fold works, nor do I see how one can rigorously prove that a 2-to-1 fold is impossible. Come to think of it, I don't even know how to state the problem as a mathematical one.
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