## Combinations of mirror symmetries

We now create rosettes with combinations of the two mirror symmetries. We can put them in “parallel” or in “series”.

In “parallel” means that the rosette has both symmetries at the same time and thus the mapping functions have to obey to the conditions f(r,φ)=f(1/r,φ)=f(r,-φ)=f(1/r,-φ). They are then

and

These mirror symmetries for themselves make a rather “static” image of four-fold symmetry. To make this effect stronger I also used a four-fold rotational symmetry with p=4 to get this:

Here I used my photo of a fly on a Muscari flower. For the X-function I put a simple cos(4φ) term and for the Y-function a cos(8φ) term.

In “series” means that we use both mirrors one after the other. This gives a rotation by 180 degrees with the center at the intersections of the two mirror axis. The image should have this symmetry which we can write as f(r,φ)=f(1/r,φ). Thus the mapping functions are

and

Because of the dynamical nature of this added rotation symmetry I preferred a rosette with 5-fold rotational symmetry. My photo of a goldsmith beetle gave this

Although this is a rather abstract ornament you can see details of the green back of the beetle and the rose flower.

I am happy that I can do this work on symmetries and create images but I cannot forget that there are many children, women and men in danger. We could help them by supporting Doctors Without Borders for example. During the next days you can get a lot of games for a small donation from  https://www.humblebundle.com/freedom.

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