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The image https://commons.wikimedia.org/wiki/File:Ball_of_yarn_10.jpg shows a ball of yarn. The spherical ball has radius \(R\) and volume $4 \pi R ^3 /3 $. The yarn itself has radius \(r\).

How long will the yarn be on average? The length $L$ is surely smaller than $V/(\pi r^2)$. But I have no idea how to estimate an actual average length. Is there a way to do this?

You want a probability distribution on the various ways of tying up yarn. I doubt that would be a very interesting calculation.

I find this VERY interesting. That is why I asked. I'd pay 50 Euros for the solution.

What way do you wire it exactly to get a quasi sphere?

Many puzzles follow the option. Consider that R and the sphere scale but r is constant and that the poles must always drift.

Phisically speaking, it is not sure that the energy equilibrium coincide with the topological optimum choosen ; you can optimize for the speed the radius R grows, for the density or for the quasi-sphere quality.

Definitively a question for the Chat :)

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