What is the shortest length of string that suffices to hang a unit-radius ball $B$?
This question is related to an earlier MO question, but I think different.
- Assume that the ball is frictionless. Perhaps $B$ is a billiard ball, and the string is nylon thread.
- Start under the restriction that you are not permitted to cut the string.
- But you may tie knots, whose total length $\epsilon$ is negligible (or we can take the infimum of all lengths).
- Let $h$ be the distance from the topmost hanging point to the north pole of $B$.
Here is a possible solution of length $L = 3\pi + h$.
The green portion is $h$. The blue forms one loop from north pole $N$ through south pole $S$ and back up to $N$, at which point it is tied and then descends again to tie at $S$. I am a bit uncertain if this $120^\circ$ partitioning could be maintained without friction. Just one great circle of length $2\pi$ would leave hemispheres exposed, allowing the ball to fall out under the slightest perturbation.
Variations are obtained by altering the assumptions above. Suppose there is friction $\mu$. Perhaps $B$ is a tennis ball, and the string is twine. Maybe then a type of spiral shorter than $3\pi$ would work? Allowing cutting of the string may help. Maybe then one could fashion a bird's nest into which $B$ nestles, achieving a length closer to $L = 2\pi + h$? Tying a knot above the north pole of $B$ could conceivably help, in which case the length of $h$ might play a role.
Any ideas would be welcomed, including sharpenings of the problem specification. I am especially interested to hear of a provably optimal solution under any variation.
Addendum (28Oct10). Here is a depiction of Scott's suggestion in the comments:
Addendum (10Feb11). Martin Demaine at MIT contacted me to inform me that this question was asked and answered long ago: H.T. Croft wrote a paper, "A Net to Hold a Sphere," J. London Math. Soc., 39, (1964) pp.1-4. (PDF here.) He credits the problem and solution to A.S. Besicovitch in a paper from 1957, same title, Math. Gaz. XLI, pp. 106-7. Here is Croft's first sentence:
has shown: if a net of inextensible string encloses a sphere of unit radius in such a way that the sphere cannot slip out, then the length of the string is strictly greater than $3 \pi$, and this is false with any greater constant replacing $3 \pi$.
This result accords with the answers below, by Scott and drvitek.