Side note: I tend to find these people very eccentric, I sorta feel like they can see the matrix 🙂 I am absolutely fascinated by this but completely lost 🙂
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I can’t compete with the 3Blue1Brown video linked, because that guy is basically the next coming of (Maths) Jesus, but it’s still a bit more than ELI5 level. My crack at a true ELI5 explanation is:
Imagine a square. Now imagine the lines making up the edge of the square are themselves made of lots of small squares. Now, for each of the small squares, imagine their lines are made up of even smaller squares. And so on, and so on, forever. No matter how closely you zoom in on a square, you will still see essentially the same image – a square, whose lines are made of squares, whose lines are made of squares, etc. It has “infinite complexity” at its edge.
Being precise, what I just described is not a real fractal, but it illustrates the idea on a visual level – this idea of infinitely repeating complexity at the edge. For a real square-based fractal, you can look up Sierpinski squares. This is an example of a “self-similar” fractal, where its pattern repeats.
However, there are other types of fractals which are not self-similar. In this case, although what you see as you zoom in changes and doesnt repeat endlessly, the point is that the level of complexity doesn’t change as you zoom – often talked about as a sort of “roughness”. No matter how much you zoom in, you will never see a perfectly smooth line defining what we intuitively think of as the “true edge, that must exist at some point” – it will *always* have ever smaller bumps or deformations at the edge.
We can write down or “define” these fractals as mathematical entities or expressions. They have some really fascinating properties and can behave very non-intuitively. For example, what happens if I double the length of every line in a fractal? In a normal shape like a square or hexagon, it would e.g. double the the perimeter of the shape, or quadruple the area of the shape… but that actually isn’t the case necessarily with a fractal! Indeed, even starting to understand what it would even *mean* to “double every line” in a fractal is not trivial at all.
So, that’s cool, but what is the point? Well, as it turns out, there are some real-world objects and problems, like the coastline example talked about in the comments, which exhibit similar infinitely-complex behaviour at the edge, so it turns out fractals are quite useful mathematical entities for analysing and understanding certain problems. *If we can understand fractals, we can understand the real-life things that behave like them.*
And that is basically what maths does. It creates interesting entities and examines how they behave under certain conditions. In applied maths, they then try to use these theoretical versions to model and predict the world around us.
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