Consider the tasks of arranging 12, 42, 162, 642,... http://oeis.org/A122973 points evenly on the surface of a sphere. The working assumption is that the solutions will be icosahedron plus triangular arrangement of points on each face, with the linear density doubling each time. Will the solutions for smaller numbers exactly coincide onto subsets of larger ones?
Maximize the minimum distance between any pair of points. Equivalently, packing circles or spheres on the surface of a sphere. A problem from the geodesic mesh.
20*(3/5)=12
20*(3/5+3*(1)/2)=42
20*(3/5+3*(3)/2+2..1) 12+90+60=162
20*(3/5+3*(7)/2+6..1=21)=12+210+420=642
20*(3/5+3*2/2+1)=12+60+20=92
20*(3/5+3*4/2+3..1)=12+120+120=252
12,42,92,162,252
http://oeis.org/A005901
Possibly relevant: An Icosahedron-based Method for Pixelizing the Celestial Sphere, MAX TEGMARK
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