From left to right: p2, p2gg and pg 9-gon packings with densities 0.9009730, 0.8998910,  0.8981235 respectively.

From left to right: 9-gon close up, 3-gon, 9-gon, 15-gon, 21-gon, 27-gon

From left to right: 5-gon close up, 5-gon, 7-gon, 19-gon, 23-gon

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.

 

 

 

TDAgIGOpresentation.pdf

I could get the Cairo tiling packing irregular pentagons in p4g ( http://milotorda.net/index.php/cairo-pentagonal-tiling/ ) but I could get it by packing irregular pentagons in p4. The output density is 0.999999963353343.

Left: Found configuration; Right: Zoom of the configuration

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

I’ve been trying to get a better solution to the heptagon packing in P2 than previously( http://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

An experiment if I could get the a tiling with a random triangle with the hIGO. The triangle is constructed via 3 randomly picked points on an unit circle. The density of the packing is 0.999999961042656.

Left: Found configuration; Right: Zoom of the configuration

An experiment if I could get the Kisrhombille tiling with the hIGO. The density of the packing is 0.999999998714671.

Left: Found configuration; Right: Zoom of the configuration