I am missing something here, Warren. How can rolling around a tiny "foobar" rotate the sphere by pi around any axis whatever? Perhaps you are envisioning rolling the sphere around the "foobar" many times, but I am pretty sure that Dan would want you to count each repetition as additional path length. On Mon, Jul 28, 2014 at 3:53 PM, Warren D Smith <warren.wds@gmail.com> wrote:
From Dan Asimov: Suppose we roll a unit-radius ball without slipping along a closed curve C on the xy-plane in 3-space. This will have the net effect of applying some rotation to the ball. For instance, if C is an equilateral triangle of side-length = ?, the net rotation will switch the N and S poles of the ball.
--WDS: I presume by "?" you meant "pi." You should write "pi" not use some unsupported non-ASCII character.
QUESTION: What is the shortest closed curve C that also switches the ball's poles? (Or at least, what is the inf of the lengths of all closed curves C that switch the poles?)
--WDS: Line segment of length pi/2, same line segment backwards, total length is pi and curve is "closed." "But that doesn't work!" you cry. "It'll just roll sphere back to its start configuration!"
Well, make teeny tiny little adjustments in the neighborhood of the two corners of this degenerate polygon. Each corner is replaced by a "foobar," which is an arbitrarily short curve whose effect is to rotate the sphere by 180 degrees round a vertical (normal to the tabletop) axis. Then it'll work. Meaning of "closed" curve here is its end & start points are same.
And obviously pi is minimum possible so this is tight up to the foobar.
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