Considering the problem of putting n points in the surface of a sphere.

[Octavio]

A

A

 

[tkhunny]

Considering the problem of putting n points in the surface of a sphere.
We're intrusted in the angles between couples of these points and the sphere's center.
The problem consists in ubicate the n points trying to maximize the minimum angle between two points, whichever.
For ubicating the points we'll use spheric cordinates. Without losing the generality we'll ubicate the first point in phi=0, theta=0
We'll call alpha to the minimum angle between two points (the one we wish to maximize).
I think I have the solution since N = 2 utill N = 7 with high precision.
I'll post the solutions obtained since n = 2 till 7 and I wish you to post better alternative solutions, and new ones for n > 7.
The proposed solutions shall specify N, phi and theta of each one of the N points and alpha (the minimum angle)
N = 2
Phi1 = 0, Theta1 = 0;
Phi2 = 0, Theta2 = 180;
Alpha = 180.
N = 3
Phi1 = 0, Theta1 = 0;
Phi2 = 0, Theta2 = 120;
Phi3 = 180, Theta3 = 120;
Alpha = 120;
N= 4
Phi1 = 0, Theta1 = 0;
Phi2 = 0, Theta2 = 109,5;
Phi3 = 120, Theta3 = 109,5;
Phi4 = 240, Theta4 = 109,5;
Alpha = 109,5
N = 5
Phi1 = 0, Theta1 = 0;
Phi2 = 0, Theta2 = 90;
Phi3 = 120, Theta3 = 90;
Phi4 = 240, Theta4 = 90;
Phi5 = 0, Theta5 = 180;
Alpha = 90
For N = 5 the configurations can be different but alpha is ever 90. We choose the most elegant.
N = 6
Phi1 = 0, Theta1 = 0;
Phi2 = 0, Theta2 = 90;
Phi3 = 90, Theta3 = 90;
Phi4 = 180, Theta4 = 90;
Phi5 = 270, Theta5 = 90;
Phi6 = 0, Theta6 = 180
For N = 6 alpha = 90;
N = 7
Phi1 = 0, Theta1 = 0;
Phi2 = 0, Theta2 = 77.9;
Phi3 = 120, Theta3 = 77.9;
Phi4 = 240, Theta4 = 77.9;
Phi5 = 60, Theta5 = 133.5;
Phi6 = 180, Theta6 = 133.5;
Phi7 = 300, Theta7 = 133.5;
Alpha = 77.9

That's a lot of not English. Can you provide a better translation?

 

[Dr.Peterson]

I think you mean this, in more standard English:

Considering the problem of putting n points on the surface of a sphere.
We're interested in the angles between pairs of these points relative to the sphere's center.
The problem consists in locating the n points, trying to maximize the minimum angle between any two points.
For locating the points we'll use spherical coordinates. Without loss of generality we'll locate the first point at phi=0, theta=0.
We'll define alpha as the minimum angle between two points (which we wish to maximize).
I think I have the solution for N = 2 to N = 7 with high precision.
I'll post the solutions obtained from n = 2 to 7 and I wish you to post better alternative solutions, and new ones for n > 7.
The proposed solutions shall specify N, phi and theta of each one of the N points and alpha (the minimum angle).
​

But the bigger issue is that this isn't really what this site is for. We're here to help people learn to do math, not to hold competitions.

Furthermore, this problem is well-known, though not solved for all N. See these pages (found by searching for "maximum spacing of points on a sphere" and subsequent searches):

http://mathworld.wolfram.com/SphericalCode.html (defines and names the problem)
https://www.maths.unsw.edu.au/about/distributing-points-sphere (packing with spherical caps)
http://neilsloane.com/packings/ (list for 3 dimensions agrees with your alphas)