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?

Vrbljanovwu

Vrbljanovwu

Answered question

2022-09-07

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?

Answer & Explanation

Alannah Hanson

Alannah Hanson

Beginner2022-09-08Added 11 answers

Step 1
Fix 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 2
Since 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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