On Thursday 08 December 2005 18:33, dasimov@earthlink.net wrote:

Gareth quotes from MathWorld: << Surprisingly, the circular disk is not the least economical region for packing the plane. The "worst" packing shape is not known, but among centrally symmetric plane regions, the conjectured candidate is the so-called smoothed octagon.

(I'll assume this is meant to be among all *convex* centrally symmetric plane regions. Otherwise, we could use a huge figure-eight painted with a thin brush, with symmetrical, thinner, channels removed at the two most-distant points. Such can have arbitrarily low best packing density.)

Yes. Or even, more simply, a huge circle painted with a thin brush.

OK, but what about the problem where the only restrictions on the planar set is that it's connected and convex? Does anyone know the sup packing density for each of the regular polygons? How about just the pentagon and heptagon? Is this number monotone decreasing from the hexagon onward?

I have no idea. I wouldn't be surprised to see some odd/even zigzagging (odd-sided polygons packing less well), or indeed similar things for multiples of 3. -- g