It’s everywhere!
Fractals are ubiquitous in nature and society. Think of it: a big object which is made up of many small objects similar to the big object - sound like anything? (Say the United Nations, made up of countries like the United States, made up states like New York, made up of counties like New York, made up of cities like New York, made up of boroughs, made up New Yorkers. The organizational body of the UN needs to secure scarce resources and quality of life for its members, as does each New Yorker for his or her family, etc....)
The lesson to learn is that scale invariance - fractality - is not an esoteric concept filling the rarified space of the academy's upper echelons, nor is it some exotic thing found only in extreme situations - it's just a reasonable idea that appears often in the world. Recognizing it helps us understand the nature of things. And knowing how to understand it inspires us to think of some clever ways to solve problems!
Another lesson to learn is that even things which seem to be very different may, in fact, share common features at the most basic level. In other words, science is a fractal! So is society!
Here are a few famous examples of fractals in the world. This is by no means a complete list. For more information, check out the writings of Benoit Mandelbrot, who greatly advanced the understanding of fractals and their huge presence.
Fractals in nature...
- Second-order phase transitions (critical opalescence, ferromagnetic/paramagnetic phase transition, etc.)
- Properties of subatomic particles (particle physics)
- Shapes, patterns, and processes in biology (ferns, DNA, tree bark, tree branching, evolution, etc. ...)
- Snowflakes
- Lightning,
- Mountains, coastlines
- Cracks (in geology, etc.)
Where it happens in technology and society...
To clarify, we mean where the concept of scale invariance is applied directly to some description of a manmade object or to society; certainly, physics still permeates....
- Economics, finance
- Networks: Internet traffic (see the image to the right), etc.
- Industrial processing
- Architecture, music, art
- Cracks (in failure of jet engine turbine blades, wings, bridges, etc.)
Thinking like engineers...
Fractals appear in nature and society, but can we start with fractal-like things to do something useful? Sure can! Here are two examples:
Data compression (image compression being one example, etc.)
If a fractal is just a simple basic block that gets repeated over and over again, then the only things you need to know about the fractal are what the building block is and how it gets repeated. That's a really efficient way to store a picture of a fractal – much less information than the whole picture. Can we use this idea to store pictures of things which are not fractals? YES! Maybe a picture of something is not a fractal, but something about the picture is fractal-like. If you write a compute program that looks for these fractal-like elements and then captures only the essential information – what the building block is and how it can can be repeated to (approximately) reproduce the original image, then you get a huge savings in how much space you need to store the picture. Inventions like this are part of the reason why we can have multimedia on the internet today.
Recursion in computer science.
Say you're looking through your home for your favorite pencil – what do you do? Open the door, look around, see something of interest: a drawer. Open the drawer, look around, see something of interest: a pencil case. Open the pencil case look around – find your pencil! Close the pencil case, close the drawer, close the room.
That's an example of a recursive algorithm to solve the search for your pencil. You break up the problem into lots of smaller and smaller pieces, but do the same thing at every level. Lots of problems in computer science can be solved using recursion – it's one of the most useful tools a programmer has. And at its heart is a fractal structure!