Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Solution 1:

The Fibonacci lattice is not the best way to evenly distribute points on a sphere. The problem of distributing N points evenly on a unit sphere is only known for specific N.

Moreover, the vertices of Platonic solids are not always optimal. This is succinctly described on the Wolfram Mathworld site:

“For two points, the points should be at opposite ends of a diameter. For four points, they should be placed at the polyhedron vertices of an inscribed regular tetrahedron. There is no unique best solution for five points since the distance cannot be reduced below that for six points. For six points, they should be placed at the polyhedron vertices of an inscribed regular octahedron. For seven points, the best solution is four equilateral spherical triangles with angles of 80 degrees. For eight points, the best dispersal is not the polyhedron vertices of the inscribed cube, but of a square antiprism with equal polyhedron edges. The solution for nine points is eight equilateral spherical triangles with angles of arcos(1/4). For 12 points, the solution is an inscribed regular icosahedron.”

There are many approximate solvers for this ( SpherePoints[] and Offset Lattice ). The Fibonacci Spiral is easy to program, but is not optimal.