We know that densest p1 configurations are lower or equal to general densest configurations. According to https://milotorda.net/index.php/packings/ p2, p2gg and pg densest packings of regular convex polygons coincide except for non centrally symmetric n-gons with a 3-fold rotational symmetry. Given densest pg packing configuration it is possible to construct p2 and p2gg packings with exactly same packing density as pg only by taking the same pg configuration and applying p2 and p2gg symmetry operations. This would mean that for regular polygons densest pg configurations are  lower or equal to densest p2gg and p2 configurations.

Below are p2gg and p2 packings of 9-gon, 21-gon and 39-gon based on respective densest pg packing configurations, although densest p2 and p2gg packings of 9, 21 and 39 gons are different.  From left to right: pg, p2gg and p2

9-gon with density 0.8986088

21-gon with density 0.9052376

39-gon with density 0.9064117

p1 with packing density 0.9743588

I found somewhat similar configuration of real pentacene 2D crystal here: https://pubs.rsc.org/en/content/articlelanding/2008/CC/b805788e

p2 with packing density 0.9743588

 

pg with packing density 0.9715909

cm with packing density 0.9499999

Here is a similar configuration of a real pentacene 2D crystal: https://aip.scitation.org/doi/abs/10.1063/1.1760076

c2mm with packing density 0.9499999

p2gg with packing density 0.949999

p2mm with packing density 0.949999

p2mg with packing density 0.9487218

p4 with packing density 0.8212764

From left to right:

	33-gon	39-gon	45-gon	51-gon
Density	0.9064438	0.9065732	0.9066543	0.9067085

 

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 ( https://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