25 Jul
2019
25 Jul
'19
8:25 a.m.
One can prove that the expected distance from a random point on the surface of a sphere to the equatorial plane is half the radius. Assuming we could rephrase this claim in a form that Archimedes would recognize, how would he have proved it? As an example of the kind of proof I would like to see, consider the proposition that the expected distance from a random point in a disk to the boundary of the disk is 1/3 of the radius. One can prove this using the formula for the volume of a cone. (I came up with this myself but I’m sure others have too.) Further examples of the kind of proof I have in mind are Archimedes’ determination of the surface area and volume of the sphere. Jim Propp