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.
Return to minimal polynomials page.