Fundamental regions of densest p2, p2gg, pg and p3 packings

From left to right 3-gon, 5-gon, 6-gon, 9-gon, 10-gon and disc. Red lines denote the boundary of fundamental regions of respective configurations.

p2

p2gg

pg

p3

Non-isometric densest P2MG configurations

I discovered that n-gons with a 2-fold but no 3-fold or 4-fold rotational symmetry have at least two non-isometric densest P2MG configurations. Below is an example of two non-isometric P2MG packings of a 10-gon with the same approximate density of 0.8372213.

Projected n-gon plane group packing orderings

We can use the packing density table to define a plane group ordering according to densest n-gon packings for each n-gon. Below is a visualization of this ordering. For given n-gon a color is assigned for each plane group based on the density of respective densest plane group packing.

Based on these orderings we can tell the densest plane group packing ordering for an arbitrary n-gon and it’s symmetries. Below are the projected orderings of densest plane group packings of n-gons for n = 5 : 42

Pentagon densest P2MM packing to PM and C2MM packings

Given densest P2MM packing of a regular convex polygon it possible to convert the configuration to a PM and C2MM packing of exatly same density. For n-gons without central symmetry this these C2MM packings are not optimal and the PM packings are not the same as those obtained using etropic trust region packing algorithm although they have the same approximate density.

Here is an example in the pentagon instance. PM and C2MM packings constructed based on densest P2MM packing with density 0.6909830 . The PM packing is different  than the one obtained by entropic trust region https://milotorda.net/wp-content/uploads/2022/06/paperPMgon5-eps-converted-to.pdf and the C2MM has lower density than the one obtained by the entropic trust region https://milotorda.net/index.php/packings/ From left to righ: P2MM, PM and C2MM.

Octagon densest PM packing to densest P2MM, C2MM and P4GM packings

Given densest PM packing of an a n-gon with a 4-fold rotational symmetry it possible to construct P2MM, C2MM and P4GM packings in https://milotorda.net/index.php/packings/

Here is an example for an octagon. P2MM, C2MM and P4GM packings based on densest PM packing with exactly same density 0.8284271. From left to right: PM, P2MM, C2MM and P4GM

CM to P2MG packings

Same thing as for the PG group (https://milotorda.net/index.php/pg-packings/) can be done for the CM group. At least for regular convex polygons. That is, given a CM packing one construct a P2MG packing with exactly same density, fairly easily. For most of regular polygons CM a P2MG packings have the same densities. The exceptions are regular convex polygons with 12k-1 and 12k+1 rotational symmetries. See:  https://milotorda.net/index.php/packings/

Below are some examples of densest CM packings converted to P2MG packings. The P2MG packings are not the densest P2MG packings in these instances. From left to right: CM and P2MG

11-gon with density 0.8279530

13-gon with density 0.8321287

23-gon with density 0.8386501

PG packings of non centrally symmetric polygons with 3-fold rotational symmetry

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

Densest plane group packings of a pentacene representation

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

More 3-fold p2 packings

From left to right:

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