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Given a unit sphere (radius 1), I would like to know how many cones I can pack into this unit sphere. Restrictions: The top of the cone needs to be in the center of origin. The bottom of the cone needs to form a circle on the unit sphere.

I have found a related question, but with a cube: Packing space by cones: Translates best?

I have also tried to find an upper bound myself by performing the following calculation:

Surface of the projection of the base of the cone on the unit sphere:
$$2 pi r^2(1 + sin(theta) pm cos(theta))$$

Surface of the unit sphere: $4 pi r^2$

Now, a (very high) upper bound would be:

$$frac{2 pi r^2(1 + sin(theta) pm cos(theta))}{4 pi r^2}$$

This however does not take into account the restrictions of the shapes, so the actual number will likely be much lower.

Question 1: What would be a closer upper bound

Question 2: If an example value is easier, what would be a realistic number of cones given $theta = 5^{circ}$

This is the problem of finding spherical codes. Putatively optimal solutions can be found at Neil Sloane's website.

For an upper bound, there's $dleqsqrt{4-csc^2[frac{πn}{6(n-2)}]}$, where $d$ is the 3d distance between some two points.

A good reference for volumetric arguments for the maximum number of 'cones' or spherical 'caps' that one can fit, is a series of papers by Jon Hamkins. The density of a packing of these caps can be at most $frac{pi}{2sqrt{3}}$, (this being known as the Fejes Tóth bound) and in the minimal distance between centers of a packing, $d$, the density is bounded above by $frac{pi}{2sqrt{3}}-O(d^2)$.

This follows from a bound on the maximum size of a code on the sphere with pairwise distances at least $d$,

$$M(3,d)leq 2left(1-frac{pi}{6cot^{-1}sqrt{3-d^2}}right)^{-1}$$

taken from chapter three of Hamkin's thesis here (being another form of Tóth's bound).

For specific parameters, the problem can be pretty intricate. One of the best references has already been mentioned in Bullet's answer and is Sloane's webpage. Codes on Euclidean Spheres, by Ericson and Zinoviev is worth checking out as well as a reference on these specific packings (and is slightly outdated), along with Sphere Packings, Lattices and Groups by Conway and Sloane.