formal definition of "fractal" or standardized categories?

Question

fractals are many decades old and come up in a wide variety of contexts and can be generated in so many different ways. however, a formal definition of fractal seems really slippery/ difficult. are there any authorities/ experts who analyze this question and try to come up with formal or informal definitions/ categories more than a sentence long, and maybe attempting more formal definitions? or maybe some attempt at general categorization? (note that a naive attempt to define fractals by their fractal dimension might not seem to succeed, because wouldnt it also come up with fractal dimensions of pure randomness/ noise? etc)

Answer 1

There's a couple of things going on here.

Deterministic Fractals:

These are fractals constructed in a algorithmic way.

In the above example, you can clearly see what makes the fractal self-similar. It's also readily apparent that it has a Fractal Dimension different from the topological dimension. These fractals, are created by defining a self-similarity relation, that's why it's no surprise that they are self-similar, they are quite literally defined to be so!

Non-Deterministic Fractals

You can clearly see that the Mandelbrot Set boundary depicted above is a fractal, the boundary is quite complicated! However, what you can't place is exactly how the boundary is self-similar. This is because there is inherent randomness in the process of making the fractal. Although, the process itself is deterministic, inherent properties of numbers prohibit a simple formula for the boundary. It's much like how we can deterministically find primes, and find their distributions, yet still the primes themselves have an element of randomness. Perhaps most interesting, is the fact that the Fractal Dimension of the boundary is $2$.

So it's really the Non-Deterministic fractals that make for a hard time. This also includes fractals with a deterministic creation, but with non-obvious self-similarities.

This is a deterministic fractal, but the actual nature of it's self-similarity is less than obvious. And of course, that doesn't even scratch the surface of what fractals are in reality/nature. For instance there are DLAs, Hofstadter's Butterfly, and so many other things that are fractal, but hard to point out why.

What is a Fractal?

Michael Barnsley presents a definition in Fractals Everywhere. He gives a formal treatment on page 356. Informally, it's a set that is invariant under a certain transformation.

Others say that a fractal's Haussdorff dimension must be strictly greater than it's topological dimension. I believe Falconer discusses that in a book here.

Google says, "a curve or geometric figure, each part of which has the same statistical character as the whole". Pretty much what the average person thinks.

Personally, I couldn't care less about using notions of self-similarity and fractal dimensions to define what a fractal is. In that way, I'm closer to the statistical point of view. In my opinion, a function is fractal if and only if it is bounded, continuous, and nowhere-differentiable in some region $R$. This is closer to the pathological function view of what a fractal is. Surprisingly relevant in the Path Integral Formulation of Quantum Mechanics.

With respect to your objection about random phenomena, yes they are fractal. A white noise plot is fractal by 2 out of the three definitions discussed above. However, if it helps, my definition yields it as not a fractal.

Answer 2

As far as I know (I've heard so many years ago), fractal is supposed to be defined as a set (in Euclidean space, I guess it could be extended to some other kinds of spaces) whose Hausdorf dimension (= fractal dimension) is grater (not equal) than its topological dimension (it's never smaller).

Fractals are not necessarily self-similar at all, but it seems that they have to be (non-formally speaking) infinitely detailed, infinitely intricate, because something in its structure has to (when scaled up) make it increase more than its topological dimension would suggest, e.g. Koch's snowflake has topological dimensiopn 1 (it's a line) but scale it to factor 3 and it will be 4 times bigger. Such things are easy to see with self-similar fractals.

Also, self-similar set is not necessarily a fractal. Let's take a square. Cut it into 4 little squares. You've just divide it into pieces that are similar to the whole set. So it's self similar. (Or a line segment, just cut it into 2 or more pieces.) Its Hausdorf dimension is 2, just like it's topological dimension. So Google's definition is just a whole load of ...

Of course, how do you construct or define a fractal that is not self-similar? You can imagine constructing Koch's snowflake by adding a spike to a each line segment at each step. And with a different self-similar fractal, you do something else (into infinity, to make it infinitely intricate). If you repeat the same process, you will get a self-similar structure. So you need to do different things. The process has to be irregular. But if you use finitely many ways to do the next step (and just stick to irregular pattern of how you apply them, kind of like decimals of an irrational number versus rational), then again it will appear self-similar (you'll zoom in and keep seeing familiar patters that will remind you of what you've seen before). I could go on and on, but I'll let you ("you" = "whoever reads this at the time") have a good think about it yourself.

For the reason briefly explained in the last paragraph, the fractals studied happen to be mostly either self-similar or at least appear self-similar (in the sense that details seem to repeat, they kind of look like what you've seen before while zooming in). That's why they are thought of as "intricate objects that look like their parts".

About the definition of fractal function in Zach466920's answer: the problem is that not all fractals (or other sets we consider whether they are fractals) are functions. I guess Koch's snowflake is (you can describe it easily with a parametric equation, and the function used is nowhere differentiable). But what can you do with Cantor set or Sierpiński triangle?

However, the objects with Hausdorf dimension greater than their topological dimension surely deserve some name. Give them a name or not, but start studyng them and you will end up studying fractals. So it seems the only sensible way I see to formally define fractals is with Hausdorf dimension. (By all means disagree with me.) I used to think that that was the official definition in maths. But now ...