We now come to the last distinct combination of symmetries for friezes and rosettes. It uses the glide reflection and the rotation by 180 degrees resulting from two mirror symmetries of the two preceeding posts. The mapping functions have to have the symmetry relations

f(r,φ)=f(1/r,-φ)=f(1/r,φ+π/p)=f(r,-φ-π/p) for rosettes with p-fold rotational symmetry.

From f(r,φ)=f(r,-φ-π/p) we see that a mirror symmetry arises at φ=-π/(2p). This gives us

and

I found it quite difficult to get a nice image that clearly shows these symmetries. The photo of a butterfly (a red admiral) gave

This image has only three-fold rotational symmetry. Look at the pale violet flowers to discover the glide reflection. The center of the rotational symmetry is at the center of the orange shapes.

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