Cyclical or “loop” fractals?
I want to ask an intuitively conceived question about “fractal loops”. However, I don’t know the mathematical definition of a fractal perfectly, off the top of my head.
Because you can “zoom in” infinitely into fractals, I suppose the idea is there is some function specifying the data of the fractal, perhaps with some characteristic property coming from real analysis… I don’t think it would have to do with limits, since I believe a fractal can indeed be continuous at a given point. I’ll have to learn more about what mathematical property characterizes how their “information” is specified on an infinitely granular level. (It reminds me of information theory… and since the real numbers have that certain property, I think uncountability, density, or something, I wonder if this “infinite variation” of a fractal’s contours is inherently tied to the properties of real numbers?)
All I want to know is, has anyone defined a “loop fractal” where as you continue zooming in or out, you actually come back to where you started? (Not a repetition of pattern, but rather, a kind of cyclical number system in which one moves from a starting point in a direction and eventually returns to where they started.)
Let me know if it is not clear what I’m envisioning. Thanks.
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Does the Cantor set qualify?
This animation loops infinitely, and shows that you get back to where you started every time you zoom in by a factor of 3.
Show animated GIF
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