A Packing Doubt

Let us call a connected set of points 'centrally symmetric' if there is a point P such that if V is a vector joining P to a point in the set, moving -V from P yields another point in the set. P is called the center of symmetry.

Question: In 2D consider the set of all centrally symmetric regions (not necessarily convex ones). Are there such regions for which the best packing of the 2D plane with copies of it is NOT periodic? If there are, which is the region with its optimal pack giving the greatest departure in filling fraction from its best periodic pack?

If the answer to the analogous question in 3D has a negative answer, the Kepler conjecture will follow as a corollary. So, since Kepler is nontrivial, there must be such connected 3D regions for which the best space filling arrangement is aperiodic. What of 2D?

This 2D question should have a known answer. Shall record it here when I find out.

Doubt: How much does one lose by way of packing efficiency if all centrally symmetric unit shapes are constrained to have the same orientation?

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Partial Answer (April 2015)

It seems that if the rotations are not allowed, using only translations, then the best packing has to be a lattice packing. It is not clear if the rotations can improve the packing density for planar centrally symmetric sets. This is for convex 2D sets.

For non-convex centrally symmetric sets, here is an example due to Prof. Roman Karasev. It packs perfectly with rotations and rather badly without them.



Note: I saw in the Wiki article on Kepler conjecture that Gauss had proved the conjecture for regular arrangements of spheres in 3D. That could mean that for some other centrally symmetric 3D figures, the regular arrangment may not be the best. I know of no example though (not even non-convex shapes). I don't yet know of 2D convex regions for which non-lattice arrangement is better at filling the plane than the best lattice layout.