Allan, That's probably because the Archimedes-based method doesn't generalise to S^(n-1) for n != 3. If you take Archimedes's theorem about S^2 and convert it into an equivalent theorem about S^3 (by means of the Hopf fibration), then you get the statement that the squared length of the orthogonal projection of a random unit vector (in R^4) onto a fixed 2-dimensional subspace is uniformly distributed in [0, 1]. Here's an article I wrote recently on the subject: https://cp4space.wordpress.com/2019/05/24/w2-x2-w2-x2-y2-z2/ -- APG.
Sent: Thursday, July 25, 2019 at 7:10 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Geometric probability a la Archimedes
Is it really true that the z coordinate is uniformly distributed?
A really common question is, "What's the right way to generate pseudo random points on the surface of a sphere?" The most common answer, and the only one I know offhand, is, "Generate three Gaussian coordinate and normalize.". Why have I never heard, "Generate a uniform z coordinate and a uniform azimuth."?
On Thu, Jul 25, 2019, 1:47 PM Cris Moore <moore@santafe.edu> wrote:
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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