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A spherical planet of radius $R$ is inhabited by $n\ge 2$ aliens. Due to the emergence of a novel deadly virus, the aliens must practice social distancing, staying away as far as possible of each other. What is the maximum distance that can be achieved in terms of $R$ and $n$?

If Euclidean distance is used (not Geodesic), the answer for $n=2,3,4$ the answer should be $2R$, $\frac{R}{\sqrt{3}}$, and $\frac{R\sqrt{6}}{4}$ respectively.

Note: this problem is about maximum distance to the nearest neighbor, not about total distance, as wrongly marked as a duplicate.

Blue
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Momo
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  • Do we assume that the aliens can be represented as points? – Jam May 29 '20 at 16:05
  • Related – Rushabh Mehta May 29 '20 at 16:12
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    I think we can use (Question 191273) and the other pages linked from there. – Jam May 29 '20 at 16:15
  • @Jam Yes. And we also assume that the entire surface of the planet is inhabitable. – Momo May 29 '20 at 16:43
  • I have already looked at those posts, but I couldn't find a way to apply to this case. In the first post, the edges are equal. In the second one, the sum of distances is minimized/maximized, which does not seem to be the case here. – Momo May 29 '20 at 16:51
  • @Momo I'm pretty sure there is general closed form that is known. – Rushabh Mehta May 29 '20 at 17:04
  • Maybe its sufficient to consider just the $n>>1$ case where the optimal packing is close to optimal planar packing with distance $d\approx \sqrt{\frac{8}{3n}}R$. – Its_me May 29 '20 at 18:48
  • You can not assume that the minimal distance is the same for all. The right approach, as Its_me hints, is looking at it from a perspective of "packing on surface" problem. You may reach a "nice" solution also for 6, 8, 12, and 20. – Moti May 30 '20 at 01:09
  • @Raskolnikov Would the maximization of total distance also maximize the distance to the nearest neighbor? – Momo May 30 '20 at 13:37
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    @Raskolnikov I checked the answer and it does not solve the problem. Maximizing the distance to the nearest neighnor and maximizing the sum of total distances are two different things, which should yield different answers in general. The quality of this site seems to have gone down the drain. – Momo May 30 '20 at 15:03
  • This is related to the Thomson problem, which is unsolved. – Graviton May 12 '21 at 04:43
  • @DavidK It looks like it's the same problem, although the paper linked in the answer minimizes some kind of energy, not the maximum distance (which means that it deals with a different problem). However, after more searching, I found that it is called Fejes Tóth's Problem and the page linked states that it is an unsolved problem. See also Spherical Code, Mathematica seems to have a function to approximate it, so it seems a well known problem. – Momo May 12 '21 at 15:10
  • I think the question as stated is your problem, since it literally says "maximum distance" and not sum of distances (and certainly it says nothing about energy). The first answer is relevant (though a bit ambiguous since it only looks at completely symmetric cases, which apply to other questions as well). The answer with the link is indeed answering a different question and not the one that was asked; this happens sometimes. – David K May 12 '21 at 16:09
  • Thanks to your research I was able to update the community wiki answer for the other question. I think that is a benefit. – David K May 12 '21 at 16:29

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