What is the average length of threads criss-crossing a hollow sphere?Imagine a hollow sphere of radius R that has a large, random (but even) number of holes in it. The surface density of the holes is constant. Threads criss-cross the sphere at random, from one hole to another. What is the average length of these threads?It is a multiple of R - but which one?

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Alannah Hanson · Accepted Answer

Step 1Fix one point p; then the density of uniformly picked points on the surface of the unit sphere at angle   θ from o is       1    2    sin  ⁡  θ, and the distance from p to a point at angle   θ is            sin      2        ⁡    θ    +    (    1    −    cos    ⁡    θ          )      2        =      2    −    2    cos    ⁡    θ  , so the average distance is      ∫    0    π        2    −    2    cos    ⁡    θ        1    2    sin  ⁡  θ      d    θ  =      1    2        ∫          −      1        1        2    −    2    u        d    u  =      1    2              [      −              1        3            (      2      −      2      u              )                              3            2                              ]              −      1        1    =      4    3      .Step 2Since the distance scales with the radius of the sphere, for a sphere with radius R the average distance is             4      R        3  , so it seems that you win the bet.

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