Re: [math-fun] Puzzle and question about pairing up points on the torus with disjoint segments

You are of course right, Maximilian. Thanks for pointing that out. —Dan

On Tuesday/5January/2021, at 2:56 PM, M F Hasler <mhasler@dsi972.fr> wrote:

I wrote: (If points p and q of T^2 are less than the maximum possible distance of √(1/2) apart, then there is a unique shortest segment connecting them.

I think that's not true. They can also be at distance 0.5 with two distinct possibilities of connecting them with segments of equal length, e.g. when both have y=0 and one has x=0 and the other has x=0.5.

One can connect them either with the segment {(x,0) ; 0 <= x <= 0.5 } or with the distinct segment {(x,0) ; 0.5 <= x <= 1 }.

So the segment of minimal length is not unique in that case where the distance is 0.5 < 0.707 < √(1/2).

- Maximilian