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Winding yarn into a ball suggests some mathematical questions:

Yarn

  1. Under some natural model, what paths should the yarn follow to achieve the most dense ball? One model is that used by Henryk Gerlach and Heiko von der Mosel in their paper "On sphere-filling ropes" arXiv:1005.4609v1 (math.GT). (This is the same model I suggested in a MO question.) I expect it would be difficult to extend the optimal solutions of the above paper to the multiple layers for a ball of yarn. (Below is shown part of Fig.6 from their paper.)

Rope

  1. What paths should the yarn follow to achieve the least dense ball? Here I imagine layers forming a grid that suspends the yarn above as much empty space as possible.

  2. Random winding. Typical instructions for how to do this by hand say, "Change directions every once in a while while you are winding," or "As you wrap, slowly rotate the ball counterclockwise to keep the distribution even." What density is achieved by random winding?

I ask these questions primarily out of curiosity. Perhaps there is an analogous process (winding the interior of a golf ball or baseball?) that has been studied mathematically.

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I fixed it up a bit; some of the links went wonky in the original version. –  J. M. Nov 9 '10 at 0:43
    
J.M.: Thanks for fixing my code! I give up trying to get the numbering to sequence properly... –  Joseph O'Rourke Nov 9 '10 at 0:43
    
Yeah, it's a weird thing with the current SE engine; not sure how to fix it too. –  J. M. Nov 9 '10 at 0:51
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Isn't the least dense ball where the yarn is unrolled completely so as to be laid out straight? –  dsg Mar 20 '11 at 2:02
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2 Answers

Depends on the elastic properties of the yarn, and on how tightly it is being wrapped.

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My problem with this question is that it doesn't feel well-defined. How thick it the rope? Can it bend at arbitrary angles? How big is the sphere to be filled? In general, one would expect that as the radius gets larger and larger, or as the thickness of the yarn gets smaller and smaller, that the asymptotic amount of space filled by a maximal filling would approach 1. If the yarn has an appreciable radius, then perhaps one would expect this to be similar to filling the ball with cylinders. There, the optimal filling yields a density of $\dfrac{\sqrt{\pi}}{12}$.

Of course, one would also expect to do better than that, as that requires all cylinders to be parallel. So perhaps we would think of this as being a bound. So the density, $\rho$, is s.t. $$\dfrac { \sqrt{ \pi} } {12} \leq \rho \leq 1$$ The bound depends on the 'squishiness' of the yarn, the limits on its velocity change (i.e. it's maximum allowed curvature), and the ratio of the radii of the yarn and the ball.

Of course, this is unsatisfactory as an answer. Ultimately, I can only say that I think this is an open problem, in the sense that even if these ideas were well-defined, they have not been fully considered.

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And I add that the least dense ball is a strange question- why not just leave it empty? –  mixedmath Apr 21 '11 at 23:00
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