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.
2 answers
Yes, most of the simpler fractals are like this. For instance, in the Sierpinski gasket, each of the three corner triangles is made up of a copy of the whole.
In fact, this is a defining property of the gasket: that is, if we state that there is a figure with the property such that it consists of three copies of the whole, shrunk by 50%, and arranged with two copies next to each other, and one on top in the middle, then we must be talking about the Sierpinski triangle.
Here is another example: the Koch curve.
The Koch curve has the interesting property that there are two ways of defining it in terms of copies of the whole. You can shrink two copies by about half and place them upside-down, next to each other at a slight angle, or you can take four copies and arrange them with two flat ones on either side and two in the middle angled up.
These are called Iterated Function Systems (IFSs). Any set of two or more maps that are contractive on average will produce a IFS with the property you describe: if you zoom in on the right spot, you see a smaller copy of the whole.
The relation to number systems that you mention also exists. For instance, we can encode each "subtriangle" on the Sierpinski gasket by which of the three transformations (0,1 or 2) we need to apply to get to that triangle. Applying this recursively, can encode smaller and smaller regions of space by longer sequences of these three digits.
From this paper by yours truly.
You can see the relation to real numbers if you imagine the interval [0,1) as an IFS with two copies: the top and bottom half are both copies of the whole. Then, the corresponding codes give you a binary sequence for each diadic interval, which in the limit form the binary decimal expansions of a real number.
Interestingly, the Mandelbrot set has this property approximately. If you zoom in on the right places, you will see many "mini-mandelbrots", sometimes called "satellites".
Created by Wolfgang Beyer with the program Ultra Fractal 3. - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=322028
The larger structure of the Mandelbrot also allows for other "looping zooms", as shown in this gif.
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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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