Can real numbers be used to create fractals?

Question

Im doing a research about complex numbers used in fractals. It is clear to me that complex numbers are used to create them, like for example the Julia and Mandelbrot sets; although I was wondering if real numbers can create fractals too.
I have to start my project with a question, so I was thinking in: Why are complex numbers used in the creation of fractals? Although if you can not create a fractal with real numbers, it would be redundant to ask it in that way.
Any help is welcomed!!
Thanks in advance.

Answer 1

Fractals show up in a myriad of contexts. Mandelbrot sort of pioneered the area of fractals, and indeed the Mandelbrot set and Julia sets are defined within the context of complex geometry. But fractals began showing up much earlier than this, notably in the work of Cantor and Weierstrass. These first examples occurred within the context of real analysis and, in particular, are defined using real numbers.

As noted in the comments, probably the most widely known example of a fractal is the Cantor set. You begin with the unit interval $C_{0} = [0,1]$. You then remove the middle third and define $C_{1} = [0,\frac{1}{3}] \cup [\frac{2}{3}, 1]$. You then proceed to remove the middle third of each of these intervals - obtaining $C_{2} = [0, \frac{1}{9}]\cup [\frac{2}{9}, \frac{1}{3}] \cup [\frac{2}{3}, \frac{7}{9}] \cup [\frac{8}{9}, 1]$. The Cantor set $C$ is then defined as $$C = \bigcap_{n=1}^{\infty} C_{n}$$ One might think that eventually in this infinite intersection, we lose everything except the endpoints - but it turns out that $C$ is uncountable. The Cantor set is extremely useful for providing counterexamples in analysis, and analogues of this construction exist in higher dimensions.

Weierstrass created what we now recognize as a fractal in constructing a continuous function which is nowhere differentiable. If you're interested in seeing his construction, you can check it out here:

https://en.wikipedia.org/wiki/Weierstrass_function

Another classic example is the Koch snowflake:

https://en.wikipedia.org/wiki/Koch_snowflake

The rest of my answer may be straying a little from your original question, but if you are working on a project in fractals, you may be interested.

A more recent (and more advanced) example is that of the Laakso graph. The Laakso graph demonstrates the utility of fractals in analysis in a striking way. Without getting into too much detail, Laakso constructed a fractal which allowed him to create a metric space sharing many strong geometric properties of $\mathbb{R}^2$, yet whose behavior diverges from that of euclidean space in a pretty fundamental way.

Generally speaking, fractals show up in the theory of metric spaces. There's a very general method for producing fractals in any complete metric space. The idea of an iterated function system captures the notion of self similarity in a quite general setting. There are some great papers on iterated function systems (Hutchinson's paper is really the main reference, but it may be a bit advanced). If you want to play around with iterated function systems, here's a nice interactive tool:

http://ecademy.agnesscott.edu/~lriddle/ifs/ifs.htm