I have been trying to find fractals in some data I am working with and (using box-counting), I think I have found one. However, when I visualize the data, it looks like an exponential, which got me curious: Do exponential functions (i.e. $y<e^{-x}$) count as fractals?

It seems like they shouldn't be, but they are self-similar – cutting off the first bit and then scaling up produces the same graph – and a basic box-counting algorithm produces a fractal dimension.

Here is a Google Colab with my exploration so far.

## Best Answer

Generally... with this loose wording you will hardly get any precise answer.

I think fractal is not quite a formal math concept, it's an informal thing and allows different formalizations.

So here's one loose answer (just my basic opinion): exponential functions are not fractals, they are smooth functions which is quite the opposite of a fractal.

A fractal (in my basic understanding) is something that has this property: when you "zoom into it deeper and deeper and deeper", it never starts looking like a straight line (i.e. it keeps exhibiting a very complex graph/structure/pattern no matter how deep you "zoom into it").

Well, exponential functions are not that. When you zoom into its graph (in some small $\epsilon$ neighborhood), it starts looking like a straight line.