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
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.
this is correct. in general on my packing pages "Trivial" means there is an easy proof not attributed to a given person, "Proved by ..." means it has been proven optimal, and "Found by ..." means it is only the best currently known, with no proof known. much fame awaits those who could find such proofs, but they do tend to be hard. for example, it was only in this century that the best packing of 6 unit squares in a square was confirmed to be a square of side 3. (i proved the cases for 7, 8, 14, and 15 squares a few years earlier.) i suspect 27 unit squares in a square will be the next to be proved. http://www2.stetson.edu/~efriedma/squinsqu/s27.gif
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:
when i retire in a few years, my guess is that you will see exactly that. :) erich