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Consider the vertices of a regular polygon. They form a uniform distribution of points on the enclosing circle. The angle subtended by any two neighbors is the same. Rotation about that angle gives the exact same pattern.

How would you describe the analogous distribution of points on a sphere (like the spines of a sea urchin)? How many neighboring points on the sphere determine the pattern and what are the constraints? From the angles subtended by the minimum number of neighbors, how do you determine the total number of points on the surface of the sphere?

user1153980
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    Well, a tetrahedron would be the simplest example. A cube might work, depending on your definition of "uniform distribution". Do you mean to consider only "uniform Polyhedra"? https://en.wikipedia.org/wiki/Uniform_polyhedron There's quite a lot of stuff on wikipedia about polyhedrons with different types of symmetry, regularity and uniformity. – Adam Rubinson Dec 19 '20 at 23:23
  • Related is Thomson problem of placing N point charges on a sphere to minimise energy – QCD_IS_GOOD Dec 19 '20 at 23:28
  • Vertices of projections of semiregular polyhedrons will work, but I don't think that is a necessary condition. I had thought the distribution of charges on a sphere was an equivalent problem. Thanks for the Thomson reference. Apparently the solution of this problem is a bit involved. – user1153980 Dec 21 '20 at 12:27

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