What are fractals about?

According to Wikipedia, there are a lot of discussions on what we call a “fractal”. In math there is a quite strict definition of a fractal. While the creator of fractals, Bernoit Mandelbrot has a more loose definition of fractals. Usually, when people talk about fractals, you hear the examples like a fern-leaf, a river, your lungs, a lightning-flash, or a tree. But people might also show you the Koch snowflake, or the Sierpinski triangle. These things are in no doubt fractals. They point out “self-similarity across scales“, the basic feature of a fractal. For example, the shape of a fern are similar to the leafs of the fern and so on. A lot of patterns in nature are fractal patterns. But there is more. This review covers a lot of occurrences of fractals in human behavior and health. Healthy human behavior has shown fractal properties, (also known as pink noise) which indicates a healthy interaction of systems. Athlete rowers have more fractal properties in their rowing rhythm, people with heart problems have less fractal-like heart rhythms, and older people with a higher risk of falling show lower fractal dynamics. If you consider this, the definition of fractals becomes a little bit more broad (less theoretical), but also more useful. Furthermore, I believe that naming subjective things, events or behavior “fractal” is very useful as well. Mainly because as far as I know, we don’t have common words for several important events that describe self-similarity on different scales in one word; fractal. In this blog I will describe a few subjects which are fractal to me. I won’t describe how fractals evolve over time. But I hope you enjoy it, and I hope you agree.

 

When you zoom into the branches of a tree you get a similar image

Fractals are self-similarity
Self-similarity is the thing what everyone agree’s with. A tree its branches are self-similar to its larger shape, just like your veins in your bloodstream (see picture above). The mathematical figures also have these properties. Fractals are about self-similarity on multiple scales. But you could also argue that humans judge people using “fractal like assumptions”. Imagine you see someone with dirty clothes, in what kind of house would you except he would live? In a dirty house right? Only based on a few details we start to infer the same judgement on different scales. When people show the same behavior on different scales, you could consider this fractal. Read more about it here. This is subjective of course, and should be considered when you meet people. But you can also think of other things, like how a Title of an article coheres with it’s actual content, and the pictures within the article. Click-bait isn’t fractal because the title is misleading relative to the actual content. But you can also think about a fractal message that a company tries to convey through its commercials, and the actual behavior of this company. Does Coca Cola really spreads happiness as we see in their commercials?

Coherence is the main thing we are talking about here, and at the more levels there is coherence, the more it is fractal. But things don’t have to cohere fully, watch this video for more information about this. You can also read my blogs about the fractalness of things or on fractal people.

This video is a (short but) nice comprehensive explanation about fractals

Fractal-zooming (Spatial fractals)
When you look at the Mandelbrot set from a far you see a large shape without any real patterns. But when you start zooming, patterns emerge which you didn’t see before (watch the video below). Usually, when we talk about zooming, you just talk about looking closer at things. But with fractal zooming, I mean the phenomenon that when you zoom into things, new patterns emerge. There are many phenomena where fractal-zooming occurs. Imagine you want to clean your room, you look at your room, and think you’ll be done in 20 minutes. But when you start rubbing your floor, you’ll notice more and more small spots on the floor, you look at your drawers, inside your drawers, everything could be cleaned. What results in an almost infinite effort to clean your whole room completely. Basically, there are several levels of what “cleaning” means; from the big picture stuff (getting the clothes of the floor) to the small detailed stuff (removing the smallest spots). Cleaning is one example, but you can also think of (your) profession as a branching tree. You might have no clue what a law-officer, a janitor, cashier, or a mathematician does exactly, but when you zoom into to their activities and story-line, they can talk for hours about what their job is exactly (spatial fractal), and how their job emerged over time (temporal fractal). During small talk, a cashier will tell you that she works in the store and does basic activities. But if you really ask what she’s doing, she can tell you all about how she weighs the fruits, what the codes are for the breads, how she handles difficult customers and so on. When you start to zoom into things that you are unaware of, an infinite world of complexity emerges (spatial fractal).


This video of the Mandelbrot set shows you what I mean with fractal-zooming

Fractal perspective
One thing about fractal-zooming is very important, it’s the fact of zooming itself (also known as focusing). I believe that the one who zooms in this video is the Mandelbrot video above is “You”. You have the power to zoom wherever you want to in your world. You can zoom into cleaning if you want to, but you can also zoom into your work activities like the cashier does. The perspective is the main thing, from your perspective you are aware of the deepest problems of your closest friends, while they don’t even exist to me. You can decide where things emerge, you can zoom into the grass in the park, on your friends, your work, and wherever you zoom, new patterns and details will appear, you are the god of your own life.

Fractal relativity
When you decide whether you’re lucky, pretty, smart, or rich, you compare yourself to other people. If you look whether you are rich, we have a tendency to compare yourself with someone who is self-similar. You wouldn’t compare your prettiness with a glass or with a dog. You probably compare yourself with people from your region, or even your friends. It is harder to compare yourself with someone with a totally different appearance, because they aren’t self-similar. So its about a focus point (the thing you zoom into) and the comparison whereof our self-concept emerges. You can play with the scales on where you place your reference/comparison frame. I have written more about it in this blog.

Fractal fitting
Suppose you are jamming with your music-band. You start to sing, your drummer starts to play a nice rhythm, while your guitarist plays a nice loop. Without that much effort, you make quite a song. When this happens, all your band-members had the same self-similar thought to play his song. You had fitting minds, like the actors from Friends who fit perfectly together, or a business that has every aspect of its product perfectly covered. A system that is connected and has similar minds, they show often shows fractal behavior. They all contribute to the same thought at the same time. But why is this “fitting” fractal? Well the self-similarity is there at different levels. If you look at the music band, there are multiple levels (musicians) who show self-similar music rhythms that make an overall nice piece of music.

Fractals are about balance
Fractal branches always split in 2 or more branches. From these branches, the fractal splits again in self-similar branches. This means that when the fractal manages to replicate, it stays balanced. The fractal like shapes of tree’s and plants support them in staying balanced, because when a branch goes left, usually a branch goes right, so the center of gravity stays in the middle. You can see many Fibonacci (fractal) patterns within the growth of plants and tree’s. Furthermore, in dynamic systems you can find properties of balance and fractal structures.

Fractals are loops
Once you are in a theoretical fractal, like a fractal tree. You don’t know where you are in that fractal. You could be almost at the top, but at the same time, you could be at the very bottom. So I leave you to decide what’s the bottom and what’s the top of the fractal, because a perfect fractal is a loop. You can go on and on without ever finding the ending of the fractal. Thus, fractals are loops. Like positive positive-feedback loops, a business sells chocolates, what is nice, that makes customers buy more chocolate, what makes the business make more chocolate, and so on. Both parties reinforce each other. Both needs fitted exactly with each other, they cohered. When things are perfectly, they start to loop. A thing that is perfect, wouldn’t change its’ working tactic would it? Artists like Escher or Bach had many “loops” within their artwork, which have often fractal like shapes. Douglas Hofstadter shows many examples in his book Gödel, Escher, Bach, and in his book; I Am a Strange Loop. 

An example how our numbers could be presented fractally – Of course, this tree goes on iterating himself til infinity.


Fractals are infinite
Fractals have a lot of relationships with the infinite. A perfect fractal by itself goes on until the infinite. Like showed in the video, it can go on and on. There are multiple forms of the infinite, but the ones I know are the countably infinite, (from 1,2,3, and so on). And the infinite set of real numbers, which is uncountable (this includes the infinite numbers between the integers, like the number 0.0000000000023…, or 123.23102015643020….). I like to see the “largest” branch of a fractal as the countable infinity, while the uncountable infinity are the side branches. Through the iteration you can kind of count the fractal but I’m not sure whether this is useful or not. I just like to represent the numbers like this in my mind.

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