Various global maxima for p2 densest packings of regular pentagons

Below are outputs of my optimization algorithm for the regular pentagon packing problem. Density of all solutions is approximately 0.9213. All of the shown packings are represented by distinct plane groups but all are elements of the p2 isomorphism class. They have different lattice groups.  (In case of p2 class the plane groups are semidirect products of a lattice group and a cyclic group of order 2). Forgetting the plane groups, all the packings are the same pentagonal ice-ray structure. Thus the same approximate density.

 

 

 

Cairo pentagonal tiling

I wanted to test if the hIGO packing algorithm would find the Cairo pentagonal tiling. According to the Wikipedia page the tiling has the symmetry of the group p4g. It took me a while to realize that I can’t get the tiling by packing irregular pentagons in p4g. The motif is actually a square with a pattern inside. At least I’ve got to test packing of squares in p4g. The output density is 0.999999987587046.

Left: Found configuration; Right: Zoom of the configuration

Packing of regular heptagons in the P2 group using the hypertorus IGO revisited.

I’ve been trying to get a better solution to the heptagon packing in P2 than previously( http://www.milotorda.net/index.php/packing-of-regular-heptagons-in-the-p2-group-using-the-hypertorus-igo/  ) and I did. It just take long. The decay of the neighbourhoods has to be very slow ie. 1/1.25. The density of the structure is 0.892690680150324.

Left: Found configuration; Right: Zoom of the configuration