In some special cases such as the optimum solution for n=32 or the cube which is an intermediate solution for n=8 which is meta-stable, or the soccer ball for n=60 which is also meta-stable the solutions are the same. In most cases however there seems to be some degree of freedom in the solution. For example, for order n=8 both the geometric and the electrostatic problems have precisely 12 solutions which pairwise have similar shape. The optimum solution is a twisted cube [square anti-prism] which consists of two parallel squares one of which is rotated over an angle of 45 degrees. For all three objective functions the optimum solution has this shape but with different distances between the squares. The cube on the other hand which ranks third for both objective functions and the hexagonal double prism, which ranks seventh represent the same solution for both cases. The twisted cube is a perfect configuration having only a single class of points but it is not isotropic [three equal eigenvalues] whereas the cube is perfect and also isotropic. The double hexagonal prism is neither perfect nor isotropic, its inertia matrix having two equal eigenvalues in the horizontal plane. Most optimum solutions are similar but usually not the same for example n=12,16,24,32 are similar but somewhat different. For n=16 both problems have optimum solution consisting of two classes [12,4] and a secondary local optimum as the third solution. The four points in the second class form a regular tetrahedron one point of which we position at the north pole and one point at φ=0. The remaining 12 points are in four planes of three points each and are at different latitudes. In each plane the points are at 120 degree angles but the triangles in different planes are not aligned in any obvious way. The 12 remaining points being in the same class are a perfect configuration but not a stationary solution for 12 points. The optimum solution for n=16 for the electrostatic problem is very similar to the optimal solution to the geometric problem and both have the classification [12,4]. Each consists of a regular tetrahedron and four planes with three equally spaced points each, which are at slightly different latitudes. For n=20 the dodecahedron is a solution as expected, but not the optimal one. The optimum solution for n=32 consists of a dodecahedron (n=20) which is a meta-stable solution for n=20 and an icosahedron which is the optimal solution for n=12. The optimum solutions for n=16 for the two problems are similar in shape but numerically different. The optimum solution for n=24 also has different dimensions for all three objective functions considered. For n=24, the optimum solution for the geometric and electrostatic solutions are two stereo-isomers of the same geometric figure but with a different parameter. Kevin Brown has this figure on his web site. These four solution are perfect as well as rotation symmetric. The similarity between the solutions for different objective functions, in particular the highly regular solutions, suggests that these are feasible shapes which may have variability in form but which are otherwise independent of the particular optimization criterion enforced. It is also interesting to compare the range of the eigenvalues of the inertia matrix for similar solutions for the three objective functions. The geometric objective function seems to have the smallest range followed by the electrostatic objective and finally the arclength objective which has the largest range. For example, the optimum solution for n=8 (the twisted cube) has two equal larger eigenvalues and one smaller unequal one whose respective values are [0.638812,0.428765], [0.585624,0.560437], [0.581642,0.568670] for the arclength objective, the electrostatic and the geometric objective functions respectively. This trend is generally observed for other similar solutions. The geometric configurations are therefore considered the most regular. The top angle for the twisted cube is a fraction of [0.3850839023705000, 0.3787334261678044, 0.2821023405030303] of a right angle respectively.

Table IV compares selected solutions for the geometric and the electrostatic objective functions. I have selected two solutions for n=16 and two solutions for n=24. In all four cases the solutions are somewhat similar but not the same. For n=4 the only non-trivial solution and also the optimal solution is the regular tetrahedron which has classification [4] for both objective functions. For n=8 I show the twisted cube which has classification [8] and which is the optimal solution for both objective functions but has different shape for each. I also show the third solution in rank which is the cube [8] which is meta-stable and which is a solution for both objective functions. For n=16 I show the global optimum and the secondary optimum which is third in the ranking order by objective function in both cases. The solutions are different for both cases. For n=24 I show the global optimum which is also a perfect solution and the third perfect solution which ranks about 6900 for the geometric objective, unknown for the electrostatic problem. The optimal solutions are perfect [24] in both cases but different. The second perfect solution is 31-th in the ranking order and stationary and is not shown. The third perfect [24] solution is stationary and ranks 6923 for the geometric objective and unknown for the electrostatic and it is different for both cases. For n=32 I show the global optimum which is not perfect but [20,12] with the same solution for both objective functions, and the secondary optimum which is eighth in rank and which is [8,8,8,8] for both objective functions but with different numerical values.