Animated Fractal related Research Interests

The fractals created by this program are the limit sets of a small set of functions that are contractions of the plane composed with rotations. In each frame, I have changed these equations slightly. I am interested in answering the following about this situation:

If the functions are changed smoothly, do the fractal sets also change smoothly?
If not, does the fractal dimension change smoothly?
If the equations are changed smoothly in a loop, as the animated fractals I have displayed on this page are, can the images created be realized as the shadow of some higher dimensional fractal projected onto the plane?

A good book on fractal geometry is:

Currently, I have abandoned research in this are to do work in other areas, but someday I plan to come back. If you have any questions or comments, email me: wphooper@math.umd.edu.

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