Monday, February 05, 2018

[aamkajwr] Worst packing convex shape

What convex shape has the worst (least) packing density?  https://golem.ph.utexas.edu/category/2014/09/a_packing_pessimization_proble.html

In two dimensions, the answer is conjectured to be the regular heptagon.  Currently the best known packing of regular heptagons achieves a density of 0.89269.  However, if a denser packing of regular heptagons is discovered, in particular, one with density greater than 0.902414, then the heptagon will no longer be the worst shape: there exists a "smoothed octagon" whose highest density packing is proven to be that number.

One way to look for better packing of a regular heptagon is simulated annealing.  Or actual annealing followed by determination of crystal structure if you can create a regular heptagonal molecule.

In going from the regular hexagon to the regular heptagon, we go from the best to conjectured worst packing shape.  After the regular hexagon, do the regular polygons steadily increase in packing density approaching the packing density of a circle?

The question could easily be generalized to a sphere or other manifolds.  What convex shape of unit area requires the largest sphere to fit N copies on the sphere?