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It is well known that the trigonal planar molecule (with bond angle $\alpha=120^{\circ}$) and the famous tetrahedral (with bond angle $\alpha\approx 109.5^{\circ}$) maximizes the angle between the vectors pointing along the bonds. So the question is this:

How do we (analytically) maximize the angle between $n$ vectors in $\mathbb{R^3}$?

Or put differently:

What is the maximum distance that one of $n$ vectors in $\mathbb{R^3}$ can be from any other of the $n$ vectors, such that this is true for all $n$ vectors?

I know that for $n=\{4,6,\require{cancel} \cancel{8},12,\require{cancel} \cancel{20}\}$ one can simply make use of the Platonic solids embedded in a circle to calculate the corresponding angles/distances. However, I have not been able to find any resources for the general case. Is this problem really unsolved?

Bobson Dugnutt
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1 Answers1

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This is known as the Tammes problem. For general $n$, it is a hard problem.

According to Conway and Sloane's book Sphere packings, lattices and groups, the optimal arrangement for $n = 4, 6, 8, 12$ and $24$ are known:

  • 4 : regular tetrahedron,
  • 6 : regular octahedron,
  • 8 : square anti-prism,
  • 12: regular icosahedron,
  • 24: snub cube

The book doesn't have the answer for $n = 20$ but it points out a regular dodecahedron isn't the answer. Furthermore, it is common the best known solutions for larger values of $n$ are not the (highly symmetric) candidates.

If you can get a copy of Conway and Sloane's book, look at the references there.

The info in this book is not most uptodate. Sloane has another book on similar topic (called Spherical codes) under preparation. I think I have seen a preliminary copy of it floating around the Web. In any event, look at Sloane's site for spherical codes as a start.

Update

As of July 2015, the Tammers problem for $3 \le N \le 14$ and $24$ has been solved. Following is a short list of papers for the solutions.

  • $N = 3,4,6,12$ by L. Fejes Tóth (1943)
    Über die Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems, Jber. Deutch. Math. Verein. 53 (1943)

  • $N = 5,7,8,9$ by Schütte and van der Waerden (1951)
    Auf welcher Kugel haben 5,6,7,8 oder 9 Punkte mit Mindestabstand 1 Platz?
    Math. Ann. 123 (1951), 96-124.

  • $N = 10,11$ by Danzer (19??)
    Finite point-sets on $S^2$ with minimum distance as large as possible,
    Discr. Math., vol. 60, 1986, pp. 3-66.

  • $N = 13,14$ by Musin and A. S. Tarasov (2012,2015?)
    Enumeration of irreducible contact graphs on the sphere,
    Fundam. Prikl. Mat., 18:2 (2013), 125-145 [J. Math. Sci. 203 (2014), 837–850].

    The Tammes problem for N = 14 (2015?) (preprint on arxiv)

  • $N = 24$ by Robinson (1961)
    Arrangement of 24 circles on a sphere, Math. Ann. 144 (1961), 17-48

Edith Mooers has an survey article on Tammes's problem on 1994 summarizes all the solutions for $N \le 12$. An online copy can be found here. Together with Musin's preprint on $N = 14$, one should have a good overview on the current status of the problem.

achille hui
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    Thank you for the suggestions! I had actually briefly looked at the Tammes problem (but the wiki-page is pretty lackluster), but I didn't realize that the involved radii of the spheres were of course there to satisfy the distance demand. Any insights as to why the symmetric cases work for the $n$'s you listed? Is this connected to a specific symmetry group or something similar? Thanks again! – Bobson Dugnutt Feb 19 '16 at 16:00
  • @Lovsovs I've no idea why it works for $n = 4, 6, 12$ and I don't think symmetry is the key. As one can see, the most symmetric candidate for $n= 8$ and $20$, the cube and dodecahedron isn't the answer. – achille hui Feb 19 '16 at 16:08