Geometry question - points on a sphere

[bronto]

It's always a treat to find a new link to my site!

rppa wrote:

This is a famously hard problem. My memory says that the general solution is not known for arbitrary n. That is, there are known good solutions, but no proof that they are optimal.

Several different statements are bundled in that.

• No general solution is known, i.e. no algorithm gives the answer for arbitrary n in a finite number of steps. I would bet that no general solution is possible.
• Empirical solutions are known for many n, but only a few are proven optimal.
• There are several practical algorithms for a "good enough" pack, but their results are never optimal (unless for very small n).

 

[bronto]

The best pack of eight points is a square antiprism, not a cube. Nor is the dodecahedron optimal for 20; the solution there is harder to describe, but suffice to say its convex hull is all triangles.

For seven points, the repulsion figure (pentagonal dipyramid) has no resemblance to the best packing. I'm a bit surprised to find such a dramatic difference for such a small number.

 

[Art Vandelay]

Geometrically shooting from the hip: uniformly spaced points on a sphere would have to form a regular polyhedron, wouldn't they?

Isn't a soccer ball a counter example?

Another question: are the solutions unique (under the proper symmetry, of course)? What about in the electrical charge interpretation?

 

[bronto]

Indeed, the best packings of five and eleven points are the vertices of a square pyramid (octahedron missing one) and a gyroextended pentagonal pyramid (icosahedron missing one), neither of which you'd normally call regular.

Most optimal packings are unique, but there are exceptions. I'm not aware of any such pairs in the electron problem; for some numbers (of which the lowest is 16) there are two or more locally-stable states, but they are not equal.