What are fractals?

Question

I've seen the famous mandlebrot and julia set fractals generated by simple equations but I have no clue what they are, how they are made from a simple formula or what their use is.

Answer 1

What are fractals?

Fractals have come up as an important question two times before the invention of computers. The first time was when British map makers discovered the problem with measuring the length of Britain's coast. On a zoomed out map, the coastline was measured to be 5,000 something or other. Sorry, I've forgotten the units. But anyway, by measuring the coast on more zoomed in maps, it got to be longer, like 8,000. And by looking at really detailed maps, the coastline was over double the original. You see, the coastline of Britain that's on a map of the world doesn't have all the bay's and harbors. A map of just Britain has more of these, but not all the little coves and sounds. The closer they looked, the more detailed and longer the coastline got. Little did they know that this is a property of fractals. (A finite area, aka Britain, being bounded by an infinite line)
The second instance of pre-computer fractals was noted by the French mathematician Gaston Julia. He wondered what a complex polynomial function would look like, such as the ones named after him (in the form of z^2 + c, where c is a complex constant with real and imaginary numbers). The idea behind the formula is that you take the x and y coordinates of a point, and plug them into z in the form of x + y*i, where i is the square root of negative one, square this number, and then add c, a constant. Then plug the resulting pair of real and imaginary numbers back into z, run the equation again, and keep doing that until the result is greater than some number. The number of times you have to run the equations to get out of its "orbit" can be assigned a color and then the pixel (x,y) gets turned that color, unless those coordinates can't get out of their orbit, in which case they're made black.
Later, Benoit Mandelbrot, an employee of IBM, thought about writing a program with a formula such as, oh... maybe Z*(n)^2 + c, and then running it on one of IBM's many computers. And they eventually got some pretty pictures. Mandelbrot was the first person to get computers do the many repetitive calculations to make a fractal look good. And now you know the mathematical aspects of fractals.
The basic concept of fractals is that they contain a large degree of self similarity. This means that they usually contain little copies of themselves buried deep within the original. And the also have infinite detail. Like the costal problem, the more you zoom in on a fractal, the more detail (coastline) you get. And this keeps going on forever and ever, so you could make a pretty movie of a fractal zooming in. Or two. So far I've made a Mandelbrot Zoom (1.1 meg) and a Julia Set Zoom (784 k). Both are AVI files, but I'm planning on converting them to streaming video or something.

Answer 2

You first have to understand functions of a complex variable, as commonly expressed by w = f(z). For every complex number z on the complex plane, the value of the function w = f(z) is plugged into f(z) repeatedly, as in f(f(f(f(...(z)))). Depending on how quickly it diverges towards infinity, the original z is given a color, black for the fastest diverging value for w. If it diverges slowly or not at all, it's made white. There's lots of variations on this idea, but this is the basic idea.

Answer 3

Do some reading, its a complex concept.
One book that makes it understandable (though a little out of date) is "Chaos" by James Gleick.
Or check out "The Tao of Chaos" by Katya Walter - it combines eastern and western science very neatly.