First experiments with packing pentagons using plane groups. Genetic algorithm was used to optimize the packing density.  It seems that the  optimal packing density is approx. 0.92 ( Packings of Regular Pentagons in the Plane ).

2. Plane group: p2

Density:  0.8550

3. Plane group: pm

Density:  0.6798

4. Plane group: pg

Density:  0.8879

5. Plane group: cm

Density:  0.7450

6. Plane group: p2mm

Density:  0.6930

7. Plane group: p2mg

Density:  0.8539

8. Plane group: p2gg

Density:  0.9104

9. Plane group: c2mm

Density:  0.6908

10. Plane group: p4

Density:  0.8423

11. Plane group: p4mm

Density:  0.5406

 

12. Plane group: p4gm

Density:  0.6957

13. Plane group: p3

Density:  0.8578

14. Plane group: p3m1

Density:  0.5236

15. Plane group: p31m

Density:  0.6980

16. Plane group: p6

Density:  0.7290

17. Plane group: p6mm

Density:  0.4914