The julia set seems to have symmetries roughly corresponding to translation, rotation and scaling.
In the following image
- You can see the horizontal translation, which leaves the extremal left and right endpoints fixed is a symmetry.
- The 21-fold rotational symmetry about any one of the 'whorls' is visible also - (I think that the number 21 corresponds to the denominator of the mandelbrot bulb the julia set comes from but I don't yet know how to compute this).
- There is also a twofold rotation about the center (and any other part similar to it)
- The scales are more difficult to describe and I don't think I have found them all so I will just avoid going into detail on this unless someone would like me to do so.
Do these have a mathematical interpretation, for example as automorphisms of the Julia set in some appropriately understood sense? Or is there some other way to describe these kinds of symmetries in mathematical terms?