This reminds me of the lovely fact (mentioned before on this forum) that a random point on a sphere of radius r has a _uniformly_ random z-coordinate from +r to –r. (This is true only in 3 dimensions.) - Cris
On Jul 25, 2019, at 8:30 AM, Veit Elser <ve10@cornell.edu> wrote:
Jim,
Strogatz did quite a bit of research on this for his recent book (pre-calculus ideas). I would first look there.
-Veit
On Jul 25, 2019, at 7:25 AM, James Propp <jamespropp@gmail.com> wrote:
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.
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