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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 $\theta = 5^{\circ}$

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  • $\begingroup$ If $\theta$ becomes small, the curvature of the sphere becomes less important. So asymptotically for $\theta\rightarrow 0$ you get your upper bound multiplied by the sphere packing constant. However, I don't understand your formula for the area of the base, neither typographically nor mathematically. It does not seem to scale with $\theta^2$ for small $theta$ as it should. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Apr 12, 2019 at 8:51
  • $\begingroup$ Seems to me that $\theta$ should be listed in the restrictions. You shouldn't rely on the title of the question for relaying crucial parts of the questions. $\endgroup$
    – Acccumulation
    Apr 12, 2019 at 17:44

2 Answers 2

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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 $d\leq\sqrt{4-csc^2[\frac{πn}{6(n-2)}]}$, where $d$ is the 3d distance between some two points.

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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 $\frac{\pi}{2\sqrt{3}}$, (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 $\frac{\pi}{2\sqrt{3}}-O(d^2)$.

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

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

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.

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