I had the very same question. But on some related pages, some packings or coverings are referred to as best known and others are said to be proven the best . . . so I'm guessing that unless the word "proved" is used, they are only the best known. It's not easy to find the absolute optimum of such things with proof. One thing a torus-lover like me would like to see added if Erich decides to do so is additional best-known or proven optima for the two most symmetrical tori: a) the square torus C / Z[i] and b) the hexagonal torus C / Z[w] for w = exp(2pi*i/3) Another interesting thing would be to see graphs of the natural integer index for each case plotted against the best-known optimum. It's particularly interesting to see the "discontinuities" in such graphs. (For example, the smallest radius circle that contains 6 nonoverlapping unit circles is the same radius as for 7, but 8 takes a jump.) —Dan
On Feb 19, 2016, at 5:48 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I'm a tad suspicious: are the circle packings on Friedman's page proved to be maximal, or are they merely the best known so far? There is no link to any proofs, or key to diameters ... WFL