What exactly are fractals and why do they even have a pattern


The only thing I got about fractals is that if you keep zooming you will see the same pattern repeated, but I don’t get it. First of all, you have to zoom on the border, so you technically can’t zoom anywhere and you get the same pattern, second why is the Mandelbrot so weird? Why all those weird shapes? Couldn’t it just be something normal? Also, why you can dezoom from a Mandelbrot fractal and those weird things happen. It’s just too complicated for my little brain

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Fractals are shapes that still have detailed patterns to them no matter how far you zoom in.

They don’t have to repeat (although often they will), they just have to keep having some kind of pattern or structure at all scales. No matter how closely you look there is something going on.

Contrast this with something like a letter “s.” At the normal scale it has some interesting patterns. But if you zoom in eventually you either get parts inside the letter, parts outside, and (if you look down to the pixel level), straight lines between them. Eventually the shape gets boring.

Taking [the Mandelbrot set](https://en.wikipedia.org/wiki/File:Mandelbrot_sequence_new.gif) as an example, you can see that as we zoom in there are still complicated patterns – sometimes looking similar to the starting pattern, but not always.

> why is the Mandelbrot so weird?

The Mandelbrot set being weird is what makes it so interesting. The Mandelbrot set is defined by a very simple set of rules. Yet the shape it gives is infinitely complicated (as complicated as a shape can be). It is the textbook example of how you can get complicated, chaotic outcomes from simple processes. If it looked normal it wouldn’t be interesting and we wouldn’t care about it.

Book recommendations if you wanna understand fractals:

*The Ghost from the Grand Banks* by Arthur C.Clark

-it’s a sci-fi novel about an attempt to raise the wreck of the Titanic and turn it into a museum/amusement park, but one of the characters is really into (edit: obsessed with) fractals and it’s an interesting character development plot / study of fractals.

Weird, I know. But if you enjoy reading and you enjoy sci-fi, Arthur C. Clark is one of the best 20th century Sci Fi authors and is very passionate about science and math in his work.

Fractals are basically drawing a shape based on a formula that can be repeated infinitely such that you end up creating smaller and smaller versions of the shape.

A super simple example is drawing a circle. Then inside the circle, draw another circle that’s half the diameter. Then inside that one draw a circle that’s half that diameter, and so on. It’s not very complex, but it is a fractal.

“why do they even have a pattern”

Because if they didn’t, we just wouldn’t call it a fractal.

There are no consensus definition of what a fractal is. Mathematicians have some ideas as to what a fractal should be, but a precise definition that capture these ideas had not been found. The reason fractals have self-similarity pattern is because things that are not that are not called fractals. “something normal” are not called fractals.

You are very correct that the Mandelbrot set, islands and many other things often sold as “fractals” are actually not so; their boundaries/coastlines are! Coincidentally, I already went into some details of this on this sub yesterday: https://www.reddit.com/r/explainlikeimfive/comments/v2a8vg/comment/iasu7p0

It is correct that most fractals are self-similar, meaning that they somewhat repeat their structure at arbitrary small levels. The definition, however, is usually quite different and based around dimensionality of the object. Fractals behave under scaling not the same as one would naively expect. Quoting my own reply in a thread here:

A line segment can be turned into 2 segments of 1/2 the size; as 2^**1** = 2, it is **1**-dimensional.
A square can be cut into 4 smaller squares of 1/2 the size; as 2^**2** = 4, we say it is **2**-dimensional.
And by 2^**3** = 8, a cube has dimension **3**.
But when applied to the [Sierpinski triangle](https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Sierpinski_triangle.svg/1200px-Sierpinski_triangle.svg.png), one notes that it can be cut into 3 copies of 1/2 the size; and as 2^**1.58[…]** = 3, it has dimension **1.58[…]**, or more formally, log_2(3).

There are even examples of line-like structures that end up having dimension 2 (or 3 or more). And, maybe completely unexpectedly, the boundary/coastline of the Mandelbrot set has itself dimension 2 in this sense!