I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where
$$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), i=0,1,2, ..., n$$ ($\phi$ is the golden section) and I want to check the validity (how close it is to an absolutely equidistant distribution of points like e.g. inscribed platonic solids) of the algorithm as follows:
Let $S_n= [s_{ij}]$ be a left triangular $n \times n$ matrix which contains the distance between the $i$-th and the $j$-th point for a given $n$ for all $i > j$ which I'have calculated to be:
$$s_{ij} = \mathrm {acos} \bigg ( \sqrt {\bigg [ 1 - (2i/n-1)^2 \bigg ] \bigg [ 1 - (2j/n-1)^2 \bigg ]} + \\ \cos (\pi(\phi -1)(i-j)) \big ( 2i/n -1)(2j/n -1) \bigg )$$
I have defined a variable $E$ which represents the minimum spherical distance from each point (note that it is computed only one time for each pair of points) and I am trying to express the relative standard deviation of $E$ as a function of $n$.
I have computed some values of the relative standard deviation (CV) with SAGE for n between 10 and 45:
n | RSD
11 | 1.02609762420603
12 | 0.762321484903851
13 | 0.955047530174084
14 | 0.817050849631392
15 | 0.865552073858752
16 | 0.745372438342098
17 | 0.775660954187498
18 | 0.701857093394853
19 | 0.611822141608497
20 | 0.631861271946514
21 | 0.701092228468854
22 | 0.540483778240496
23 | 0.659987407100965
24 | 0.672830188457424
25 | 0.652372883095057
26 | 0.685352476850355
27 | 0.739996218384178
28 | 0.723356854298303
29 | 0.735561195103908
30 | 0.718544791870398
31 | 0.722160137598222
32 | 0.664300701441229
33 | 0.691263058551244
34 | 0.679994429078862
35 | 0.628303047521343
36 | 0.646751221409871
37 | 0.670926863221861
38 | 0.643799084939139
39 | 0.684100431791817
40 | 0.707904019539385
41 | 0.738481416277019
42 | 0.764314173516170
43 | 0.756728802421113
44 | 0.777566826519663
45 | 0.754029151732675
Plotting:
Makes me think that $\mathrm{CV} (n) = a_1e^{-a_2n} \cos(a_3n + a_4) + a_5 $ but this is just an estimation.
Is there a way to actually express CV in terms of $n$ (not by estimation and model fitting practices)?