The question is simple, but the answer is very complicated, and very long. Giving you a technical answer, though it may be accurate, won’t do much good as there will be so much typical ‘fractal-speak’, the jargon that not so many people understand. In high school, we learned that fractal is a never ending pattern, and fractals are built by repeating something over and over again.
Last time I checked fractals were not a part of high school final exams, still a lot of students want to know about fractals in math. I am currently working with one of the GED exam prep websites for students to create videos and some other interesting stuff not only for high school students but for the general public. Make sure you check that.
So let’s start learning about fractals, it’s so fascinating stuff!
A simple answer could be that fractals are shapes that, regardless whether you look at a fractal’s bigger or smaller part, have the same or similar, though not necessarily identical, appearance as its full shape. Take a rocky mountain for example.
You can see just how rocky it is from a distance, and up close, the rock’s surface is pretty similar. Little rocks come with similar bumpy surfaces as big rocks, just like the overall mountain.
The whole concept, or idea, of ‘self-similarity’ may be a little challenging to understand, yet it is an essential and fundamental element when it comes to understanding what fractals are. Smaller areas of a fractal shape are looking much like larger images. You can enlarge or zoom in as often as you want, but you will always see the same shapes and details, no matter how tiny the image of the full-size image.
This is what self-similarity is. Now when something is continuously so self-similar, what’s the point of zooming in, because, after all, everything looks the same. Small details look exactly like large details, so what’s the idea?
Not everything is so similar …
For fractal enthusiasts, fortunately, there’s more. A lot of types of fractals will appear wildly different when you’re zooming in. They still are self-similar, but not rigidly self-similar, and exactly this is what turns fractal exploration into something so intriguing. You will see something different every time you zoom in.
You may be surprised that, while there is so much familiarity, you’ll encounter unexpected new twists. Just one single fractal shape can always offer you something new to explore, and the further you’re zooming in, the more chances you’ll have to see something nobody has ever witnessed before. And today’s computers allow you easily to zoom, and zoom, and zoom …
How it works …
It’s not that difficult. Basic fractal techniques can be explained without needing to turn to confusing mathematical jargon and equations. You can start with giving every point on a screen a unique number. Take one number and put it into a specific formula. Then, put the outcome (the result) from the formula and put it back into that formula. Repeat this over and again, and see what will happen to the numbers that you get. Give each point a color based on what’s been happening.
That’s all. Really, that’s all there is to it. The thing is that with most formulas, this won’t result in anything interesting, but fractal creation is using formulas that allow very interesting things to happen. At times, the numbers resulting from feeding back the results into the formula, a process called iterating, will be exploding into huge numbers, that will just get bigger and bigger. These points will be colored in one way. At some other times, the numbers will get closer and closer to the original number, or ‘home in’. These will get colored differently.
The interesting thing here, and the reason why fractals are working at all, is that there are times that the tiniest change in the number that you begin with, may result in totally different outcomes as you keep on iterating that number. And you’ll also see that the boundary between numbers that ‘home in’ and numbers that will explode is very complicated and twisted: this is the shape of a fractal.
An enormous task …
If you want to calculate fractals in this way, you’re faced an enormous task. A tiny fractal image, maybe just 640 x 480, is already containing more than 300,000 points, and each of these points may be needed to be run through your fractal formula over a thousand times. This implicates that the formula must be computed over three hundred million (!) times, and note that this is just a very mild example.
Bigger images, for example, poster-size fractals) may involve over a trillion calculations. Fortunately, modern-day computers are fast and can do that job in just a few minutes, though larger fractals may take hours or days to complete, but fact is that exploring fractals never was easier than today.
And now …
As I said before, fractals are a main topic. So far, we’ve only been talking here about one specific type of fractals, the escape-time fractals, but we can find so many other types of fractals. Fact is, though, that if you dig deeper into fractals, you’ll need a lot more math to understand it. There are not so many books and publications related to fractals, and most web pages also don’t get so deep into heavy mathematics, so there’s a lot of work to be done, and a world of fascinations to explore.