convex shapes: the smoothed octagon has a packing density of about 0.902414, the smallest known for centrally-symmetric convex shapes and conjectured to be the smallest possible. (Packing densities of concave shapes such as star polygons can be arbitrarily small.)

Other packings

At the other extreme, Böröczky demonstrated that arbitrarily low density arrangements of rigidly packed circles exist.

There

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