I have heard of definitions along the line of saying that a transformation T of something with respect to a certain property p is symmetric if, after T is applied, p remains unchanged. Is there a way of refining this definition? How could you use this definition to show that fractals are symmetric? What are T and p? If something satisfies a recurrence relationship, is that a type of symmetry?
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From the comments, I am getting the impression that there is no specific definition of mathematical symmetry. Fractals are frequently described as being highly symmetric. The idea is that you can uniformly contract or expand a fractal and lay it on top of itself.