how these packings were found.

First a system of polynomial equations, arising from the equal length of many minimal edges, is solved numerically. This was done to high precision (e.g. at least 30 and sometimes as many as 100 decimal digits) using a multivariate version of Newton-
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.

Return to minimal polynomials page.