Optimal yarn balls
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.
Solution 1:
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.
Solution 2:
I think this isn't well defined yet. Wrapping a ball of yarn is not the same as filling a sphere with yarn. Case in point: your most-dense spherical picture, which would almost instantly fall apart.
I think one also has to come from at least two stand points: wrapping cylindrically, where the cylinder eventually turns so as not to destabilize, and wrapping spherically, where you attempt to wrap the ball in a three-dimensional pattern.
What plays into this? Well, first: strings have a point of contact with a string below, at which point they wrap in a circular arc of inner radius equal to the radius of the string, rotated through the differential angle. So if they cross the string below perpendicular to that string, then they have an arc radius equal to the radius of the string itself. But if they cross at a flatter angle, the arc becomes elliptical.
Second point: from support to support, a string approximates a straight line.
Third point: it matters whether the string crosses a supported string or an unsupported string. Crossing an unsupported string will crimp down the lower string, averaging energies of deflection. That lower string, in turn, may crimp down on others below, modifying previous wrappings.
Fourth point: strings inherently want to not turn their angle of wrapping, but may turn at support points, subject to friction and (possibly) sliding off their support point in a mode of failure, either partial or absolute. Partial failure is a dynamic change that does not result in the collapse of the entire ball or an entire loop. The study of partial failure of the ball of string might provide a whole interesting field in and of itself.
Fifth item: if a string is wrapped around a ball, say in a vertical loop, the ball turning on a vertical access only each time, then at the top and bottom nodes of the ball, the height of the wrapping point increases until instability results. Therefore, the ball must be turned on a non-vertical axis, resulting in a precession of the wrapping point. So talking about the optimum path may involve studying the path of the wrapping point. But the path of a wrapping point isn't a path only, it is a nodal path, with support at the nodes. This is important, because it allows the wrapping path to cross itself.
Sixth point: the wrapping path's crossings of its own path can itself create or eliminate instabilities: that must be studied too. It is the wrapping path, and the crossing of wrapping paths that affects ball density.
We could design a stable ball to have minimum density, or maximum density.
So anyhow, these are my current thoughts on the matter.