Suppose that we choose \(n\) diameters in a sphere. Each diameter intersects the sphere's surface at points, defining two hemispheres (one centered at each point.)

Now, a research paper I found (linked here: https://www.mathpages.com/home/kmath327/kmath327.htm) says:

"By choosing \(n\) diameters, we partition the surface into a certain number of subsets. Each of these subsets lies entirely inside or entirely outside the hemisphere centered on each of the \(n\) points."

I'm assuming that "subsets" refers to regions on the surface of the sphere...

But how does choosing \(n\) diameters partition the sphere into subsets? Can someone please describe this process intuitively?

Thank you so much!

jsaddern Dec 27, 2021