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Vandaag is het 22 February 2018

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Somewhere on a sphere - math puzzle

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What is the mean distance between two randomly choosen points within a sphere of radius 1?

Explanation

This problem is most easily solved by introducing spherical coordinates.

bolcoordinaten

The probability that a randomly choosen point lies in the area of the sphere with sizes dxdydz is given by

formule

The distance between two randomly choosen points formule and formule within the sphere is

formule

where formule is the angle between the points formule and formule. The mean distance between two randomly choosen points within the sphere is now given by

formule.

Using spherical coordinates and symmetry, the last formula can be simplified to

formule.

Performing the integral over the angle gives

formule.

This integral can be computed by splitting it into two parts:

formule.

Hence, the mean distance between two randomly choosen points within the sphere is 36/35 = 1,02857....

The strange thing of this problem is that it easier to solve in three dimensions than in two! More dimensions almost always implies more problems. Just try to compute the mean distance between two randomly choosen points within a circle.