N = 8: Eight particles arrange themselves into two squares on parallel planes, with the squares rotated by 45 degrees relative to each other. The 28 separations between particles come in the following four lengths:
a = 1.2876935 b = 1.8968930 c = 1.6563945 d = 1.1712477
We observe that c = d, so the two parallel square faces have edge lengths d. This configuration has E = 19.675..., as opposed to E = 22.485... for the vertices of a cube inscribed in a sphere. This shows that the vertices of Platonic solid are not necessarily
in a stable equilibrium configuration. In other words, perfect symmetry does not imply stable equilibrium.