Raphson. If this converges normally we can be nearly certain the packing exists.

At this point one can attempt to solve the system pf polynomial equations
using Groebner basis methods or by using resultants with maple or other
computer algebra system. I used maple. Only when *n*
is small can one have much hope for this to succeed. When it does
we can be sure that the packing exists. Further, we will usually
have found a univariate polynomial one of whose real roots is *u*,
the algebraic number associated with the polynomial.

In this way, or in some cases using the LLL algorithm, I have found
the following minimal polynomials for u=cos(central angle to minimal edge)
for *n*<19 and for *n*=20, 24, 38, 48, 50, 52, 120.
Thus, for example, the optimal solution for Tammes' problem for *n*=10
has *u* satisfying the polynomial below.