Winding yarn into a ball suggests some mathematical questions:
- 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.)
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.
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.