For a rosette of p-fold rotational symmetry you have an equivalent to the glide reflection symmetry of a fries. It is a rotation by an angle of π/p around its center together with a reflection or rather inversion at r=1. Thus the mapping functions have the condition f(r,φ)=f(1/r,φ+π/p), which results in

and

where I have used that cos{lp(φ+π/p)}=cos(lpφ+lπ). Using a photo of a yellow dragonfly I get this result of six-fold rotational symmetry:

Note the yellow horseshoe like shapes. The glide reflection makes that there are 12 copies, six opened towards the center and six opened outwards. Together they form a prominent star.

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