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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Well they can’t see em everywhere, they’re probably mistaking that for the fibonacci sequence and other ratio based stuff. If they were seeing fractals everywhere, they’d be seeing stuff like a strawberry with tiny little exact strawberries for seeds, which also have tiny strawberries for seeds, etc etc.
Fractals are a recursive thing you can get into with math; after repeating a problem, you get a pattern. If that pattern is drawn, like by a computer, it’ll be an infinitely zoomable image thing.
Now, the fibonacci sequence? That’s something to look up.
A fractal, by my understanding, is a self repeating pattern. This means that, as you zoom in closer you just get the same image back. So, take tree formation for an example. You zoom in to the trunk and it divides into two smaller trunks, then into two smaller trunks again, and then eventually it’s splitting into branches, then smaller branches, then even smaller branches, etc. All of those structures (the splits) as very very similar to each other, in fact they are just the same pattern repeating over and over. The same thing with our vascular system in our body. [Here](https://youtu.be/b005iHf8Z3g) is a visualization that may help!
Start by reading The Golden Section: Natures Greatest Secret by Scott Olsen. Its a breakdown of the underlying concept youre asking about. Its really short and very easy to read.
Let’s say you’re trying to measure how long a coast is. The rough estimate would be to find the end points of what you’re trying to measure, and take the distance. Of course, this won’t be accurate, because the shore isn’t quite straight – there are ins and outs. Alright, no problem, we can measure smaller distances.
The problem is, the longer you do that – the smaller the pieces you look at – the more details you’ll catch. You should probably include the cove, but what if the cove has a large rock at the shore where water doesn’t quite reach around? What with smaller rocks? Do you measure their surface? What about the sand that water goes around and into? Molecules?
The closer you look at something – even where you would’ve previously thought you have something smooth, you find that it’s made of more nonsmooth stuff. A fractal is a shape where this goes infinitely – not quite unlike most of real world. One of their most interesting properties is that they do not have a measurable length – like the coast example.
Excentric might be right. It’s hard to comment on this, because we don’t actually know what those people are talking about, but I suspect they are actually the kind of people that have a casual interest in popularized science and math, and they repeat what they heard or read somewhere. The type of people that will tell you that tomatoes are not vegetables because they heard somewhere that they are fruits Or that will swear up and down about the natural beauty of mathematics then get hilariously mad at you when you tell them about Gödel. Or that will tell you that the Fibonacci spiral is the shape of galaxies and plant leaves and snail shells.
Fractals aren’t “everywhere”. They are a mathematical construct, and most things don’t resemble them at all. The thing about fractals is that:
1. They are self similar, meaning that they contain parts that resemble the whole.
2. They contain these structures at arbitrarily small scales.
Obviously you don’t actually get arbitrarily small scales in nature at all. But some things seem to exhibit this self similarity. Certain leaves for example will grow in a way where they split and seem to grow smaller versions of themselves. Trees in general grow in a way where branches will split from the trunk and grow new branches and so forth. But, much like with the snail shells that sort of look like a logarithmic spiral, you can see that these aren’t actually fractals: those new structures aren’t actually the same as the bigger ones (and, again, they obviously don’t repeat at arbitrary sizes).
The world’s greatest [math educator](https://youtu.be/gB9n2gHsHN4). Change my mind. This deep dive on what it means to have some n dimensions and how that relates to being a fractal is hugely beneficial to not only answering your question, but also understanding more patterns that show up across mathematics.
I can’t say anything that would add to the genius that is the video linked. I can say, I have a math minor. I’ve watched this video 5 or 6 times I love it so much, and even though I understood the general overview first time, I still catch new things every time. So I can highly recommend watching multiple times.
ThreeBlueOneBrown has an excellent (https://www.youtube.com/watch?v=gB9n2gHsHN4) on the subject of fractals and how they are far more interesting than merely infinitely self-similar curves. Ultimately fractals are a way to describe curves that are not smooth. Fractals remain complicated, rough, no matter how closely you look. Self-similar shapes are a helpful example of this, but not the definition. He introduced the very intuitive box-counting method for figuring out dimension in the video.
The definition has to do with the idea of fractal dimension, itself a way of expanding the concept of dimension for non-counting numbers. The stereotypical example, the coastline of Britain, has a dimension of 1.21, meaning that you will continue to find details no matter how far you zoom. The fractal dimension reflects the rate at which new details become apparent as you zoom in.
“People who understand fractals” that’s already a false premise. Fractals are just patterns that repeat when you zoom in/out something. You must have seen pictures of fractals, so you should already have a visual understanding of what it is.
Also other comments have given good explanations, so I will stress out just a fee points: you can find plenty of repeating patterns in nature but 1) they don’t go on forever and 2) the patterns don’t repeat perfectly, so it’s real fractals in a mathematical sense.
And you will understand fractals better with visuals, so Youtube is a better place to learn about fractals.
in the most general, non-rigorous sense, fractals result from repeated application of the same rule. One way of thinking how they show up a lot irl is because most of the time every day is a lot like a repeated application of the day before.
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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