Ed wrote: << See http://www2.stetson.edu/~efriedma/cirRcir/ That's a very interesting set of solutions. Modifying it a bit will allow more asymmetry: n circles with the largest possible sum of radii packed inside a convex unit area.

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This may be related to the sausage conjecture. From a write-up on the Web, at < http://cambmathsociety.org/Math-Library1/Popular%20and%20historical/Szpiro%20G.G.%20Kepler's%20conjecture%20(Wiley,%202003)(ISBN%200471086010)(306s).pdf >: << . . . the "sausage conjecture." It is related to Kepler’s conjecture and was proposed by that Hungarian grandmaster of discrete geometry, Laszlo Fejes-Tóth. To this day it remains unsolved. To illustrate, let us observe Fumiko, a shop assistant in Tokyo who boxes up gifts for the customers in the sports department. The question she faces is, what is the best way to pack balls? Which boxing method wastes the least amount of space? For example, is it more efficient to pack four tennis balls—or Ping-Pong balls, or golf balls—into a square box, or an elongated box or a pyramidal box? How about six balls, or three, or twentyfive? We will call the long boxes sausage-boxes and all others cluster-boxes. The results are quite surprising. For up to fifty-five balls the sausage seems to represent the best box. It may be a little awkward to load such a thing into the trunk of the car, but as far as wasted space is concerned it’s the best. If you want to pack additional balls something strange happens, however. For fiftysix balls or more, clusters are better than sausages. Fejes-Tóth was so shocked and confused when he found out that he called the switch from sausages to clusters the “sausage catastrophe.” Nobody is quite sure at exactly what number of balls optimality changes from sausage to cluster. It is believed that the switch occurs somewhere between fifty and fifty-six balls. But at exactly what number this happens is still an open question. If you become bored with regular tennis you may prefer to play the game in four dimensions. As you might expect, the problem does not become any easier. As long as you play four-dimensional tennis with less than about seventy- five thousand balls, it is better to store them in sausage-boxes. For more than 375,769, cluster-boxes are best. Somewhere in between the switch takes place. Where exactly? Nobody knows. Surprisingly, in still higher dimensions the situation calms down again. Fejes-Tóth claimed that in sufficiently high dimensions, sausages are always best, no matter how many balls you want to store. This contention was proved rigorously for all dimensions greater than forty-two. But Fejes-Tóth conjectured that in any dimension higher than five, sausages are always best. As of this writing, the claim is still awaiting a proof. By the way, can you guess what happens in two dimensions? It’s real simple: clusters are always superior to sausages.5 And in one dimension, sausages are the only option. If the surface area of the box’s walls are of primary concern and not the volume, then there is another conjecture in store. The “spherical conjecture” states that the optimum, wall-minimizing shape of the box for tennis balls is roughly spherical if the dimension is large. Whether this is true, and for which dimensions the conjecture should hold, is anybody’s guess at this stage.

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--Dan Sometimes the brain has a mind of its own.