It turns out that there is associated to each packing an invariant which is an algebraic number.  This algebraic number, which we denote by u, is the cosine of the central angle to a minimal edge of the associated polyhedron.  Thus, for each n, the best packing of n circles on a sphere has associated to it a polynomial, the minimal polynomial of the packing.  This is just the minimal polynomial of the algebraic number, u, attached to the packing.  We let u denote both the algebraic number and the indeterminate of the minimal polynomial.

Why u is algebraic.
The existence question.
How the minimal polynomials were found.

Here are the minimal polynomials.  So far as I know these were not previously known and have not been published elsewhere.

I was only successful in finding such polynomials for n<19 and for n=20, 24, 38, 48, 50, 52, 120.

u=cos(central angle to minimal edge).  Thus, for example, the optimal solution for Tammes' problem for n=10 has u satisfying the polynomial named  below.


The minimum polynomial factors over Q(sqrt(2)) when n=8, 16;
the minimum polynomial factors over Q(sqrt(3)) when n=38;
the minimum polynomial factors over Q(sqrt(5)) when n=11, 12, 120;
the minimum polynomial factors over Q(9th roots of unity) when n=7.

For these same polynomial in maple format  click here .

Return to packing circles on spheres.