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 and the C2MM has lower density than the one obtained by the entropic trust region From left to righ: P2MM, PM and C2MM.