Imagine 5 unsociable people live on the surface of a tiny spherical world. Being unsociable they want to arrange themselves so the smallest distance between any two them is as big as possible. How can they do this?
If there were 6 of them they could position themselves at the vertices of a regular octahedron, as in this model [1] of a Sulphur Hexafluoride molecule, where 6 Fluorine atoms surround a single Sulphur atom. Electrostatic repulsion causes the Fluorine atoms to take up this octagonal configuration.
We can measure the distance between two points that lie on a sphere by giving the angle they make at the centre and you can see that in this case all the angles are 90°.
But now suppose one of them jumps in a spaceship and flies off the some other tiny world. Now the remaining 5 have a little more space to spread themselves out more.
Except they don't! 5 points can't do any better than 6 could and the best minimum distance that can be achieved is still 90°. Here's my attempt at a proof.
Here I showed that
If there are five points on the surface of a sphere, then no matter how they are arranged at least 4 of them lie in the same hemisphere.
I think this result may have first been published by Martin Gardner, although I don't have the reference.
You can think of this as being like the Northern hemisphere, including the Equator and the North Pole, and 2 or more points must lie on the Equator.
If all 4 points lie on the Equator the greatest possible spacing is 90°, achieved when they are situated at the corners of a square inscribed in the Equator.
Otherwise we must have at least 2 on the Equator and at least 1 in the same hemisphere but not on the Equator. The greatest possible separation between this point and the 2 on the Equator is 90°, achieved when that point lies at the North Pole. So, either way, there must be a pair of points separated by 90° or less.
We'd like the configuration to be as symmetrical as possible, so we can put 3 points at the vertices of an equilateral triangle inscribed in the Equator and the other 2 at the North and South Poles.
At this point you might wonder if there are molecules with 5 Fluorine atoms surrounding a single atom of some other element, and if so are the Fluorines in the arrangement described above. The answers are Yes and Yes. Here's a picture of a Phosphorus Pentafluoride molecule [2]
This problem of optimally spacing points on a sphere, which has always fascinated me, is known as the Tammes problem and the best configuration for 2-14 points and 24 points were found and proved between 1943 and 2015 [3]. It is a hard problem, in that it has to be solved for each number individually. I imagine we can find estimates for the general case but finding exact solutions is challenging.
[1] Wikimedia Commons public domain.
[2] Wikimedia Commons public domain.
[3] Wikipedia Tammes problem.
