How many cones does it take to cover a unit sphere, and how far away are their vertices?
This is a fun problem that I am realizing I don't have the geometry for. I was thinking about it the context of a teleporting spaceship surveying a idealized spherical planet.
- How many points would it need to survey from to get a complete map of the planet's surface?
- How far away from the planet's surface would they be?
Put more mathematically, if x is the distance from the surface of the unit sphere, f(x) = # of tangent cones required to cover the sphere with minimal overlap.
The answer to question 1 already partly exists, but it's not as trivial as it seems. Cones tangent to a sphere ultimately just describe some circle on the sphere's surface, so another way to phrase question 1 is 'how many circles of equal size does it take to cover a sphere?' It turns out this is a covering problem related to the Tammes packing problem, there's a 1991 paper on it here. Obviously the maximum number of cones you can have is functionally infinite as x approaches the limit of 0, but the minimum number of cones is 3.
As for question 2, the linked paper gives us solid figures for n circles covering a sphere, but not for every value of n. We know 2<n<15 except for n = {8,9,11,13}, and for 15 and above we don't have proofs, just conjecture. The figures for circle density on the surface of the sphere could be reverse engineered to give the vertex distance of the corresponding tangent cones, but that's where I get lost. I'd be curious what the values are though, and what they look like as a graph of f(x) as laid out earlier.
Another way to phrase the problem is "How many cones does it take to cover the unit sphere if all their vertices originate on a larger concentric sphere, and how does the number of cones change as the size of the larger sphere changes?"
I hope someone else finds this interesting, I've been fascinated by it since I concocted the problem. It's fun to find out that a thought experiment you came up with listening to an audiobook doesn't have a complete solution available in modern mathematics!