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:

Besicovitch

`[1]`

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.

metastable, which is quite apropos here! $\endgroup$ – Joseph O'Rourke Oct 27 '10 at 14:172more comments