# What is the spherical equivalent of splitting a circle into n equal segments and calculating their central angles?

So this is easy to calculate in 2 dimensions, if you have a circle represented by 3 points the angle between any two consecutive points and the spheres centre is simply $\frac{2\pi}{n}$. I basically would like to know if there is a version of the same thing in 3d for spheres.

The reason why I need to know this is because I am trying to calculate the bonding angles of different molecules. For example, methane has a tetrahedral shape and whilst I could just go look up the H-C-H bond angle, it would be better to have a general formula. This is also how I realised it was actually a math problem not just a chemistry problem.

• This looks like a though one. What do you mean exactly by "equidistant"? – Daniel Robert-Nicoud Sep 13 '17 at 16:43
• Evidently, by "equidistant" you mean that, in 2D they form a regular n-agon inscribed in the circumference. But in 3D you are limited to the five platonic solids, if you are imposing full symmetry. Otherwise you shall specify how the points are .."equidistant" – G Cab Sep 13 '17 at 16:59
• You can find the central angle of platonic solids in this link: en.wikipedia.org/wiki/Table_of_polyhedron_dihedral_angles – Dynamic Sep 13 '17 at 17:13
• @GCab That's kind of my point. A possibility is to ask about configurations minimizing potential energy for $n$ charged particles constrained on a sphere. – Daniel Robert-Nicoud Sep 13 '17 at 17:31
• that's a well known problem. see e.g this page – G Cab Sep 13 '17 at 23:44