I seem to recall that the octagon is the regular polygon with the lowest optimal packing density (sup of the density over all its planar packings where density is defined). I'm not sure whether the actual density is known, or whether there is a packing realizing the sup over all density-defined packing. I also seem to recall that there were some adjustments to the octagon that give a convex shape with an even lower packing density than the octagon. I don't know if the inf over all convex shapes S of the sup over all density-defined packings' densities is realizable by a specific shape (though I would expect so) or by a specific packing of that shape (no idea). Does anyone out there know more about this? --Dan Gene wrote: ----- I thought red octagonal stop signs are an international standard, except for "STOP" being in the local language. But, this being posted to math-fun, I would think the intended interpretation is how to fabricate stop signs while minimizing the usage of sheet metal. -----