Quote:
Originally posted by Undertoad
What's the Mandelbrot Set, you ask. Here's a quick introduction to it. It's hard math... basically, it's a mathematical function that is a fractal. What's a fractal, you ask? You ask some damn difficult questions don't you? I didn't major in math!
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I majored in math, and to be honest, that's not hard math (and from the viewpoint of a mathematician, it's not even vaguely hard math, you don't even want to know what hard math is). The level of understanding you need to comprehend the mathematics of it is about at Algebra 2 in high school. The mathematics that create these fractals have been around for centuries, but there was never any way to discover fractals until the invention of computers, doing the calculations for one image like that would take the lifetimes of thousands or maybe even millions of people.
If you want a shorter explanation of the Mandelbrot set than that link provided, I'll try to explain it as simply as I can:
First you need to understand imaginary and complex numbers. This is the stuff that was covered in Algebra 2. If you don't know what an imaginary number is, stop reading this explanation because it will just confuse you. Basically all the point of the Mandelbrot set is squaring and adding complex numbers over and over:
A complex number has a real and an imaginary part, so when make a graph, you make the X axis correspond to the real part, and the Y axis to the imaginary part. This gives you a complex number for each pixel you graph. Make sense so far?
Then you go through each pixel you want to graph, and call it's corresponding complex number C. Then you take another variable, usually called z, you run it through this repeatedly:
z = z * z + C
You just keep running it through that equation over and over and over again. For most values of C, z will go to infinity. It will just keep gettting bigger and bigger and bigger, each time you do it. That's what you'd expect from "ordinary" real numbers. If you just keep adding and squaring, of course it's just going to keep getting bigger.
But a strange thing happens when you use complex numbers. Most of them do get bigger and bigger forever. But for a small set of them, instead of getting bigger, z just hovers around zero, and it never goes off to infinity. All the numbers that do this belong to the Mandelbrot set.
Not too hard to understand, I hope. The place it really starts to get interesting is when you get to the edge of the set. It's not like there's a smooth solid border between areas that belong to the set and those that don't, it's an infinitely detailed pattern that could never have been discovered without modern computing power.
I haven't read how the guy who created the "Buddha" image chose to represent the Mandelbrot set graphically, but the "standard" way is simply to use different colors to represent how fast each point goes off to infinity.
I'm sure there are better programs out there to graph fractals, but I wrote (in a couple hours) a Mandelbrot set graph generator for a class project back in college, if anyone wants to play with it:
http://linknoid.net/software/Fractals.exe
Regarding the question of what a fractal is, now that is a harder question, and I'm not going to tackle it in length here. I've read Benoit Mandelbrot's "The Fractal Geometry of Nature" where he explains it in great detail, but to sum it up, it's something that occurs in a fractional dimension. It's
way beyond the scope of this explanation to explain what that really means, but by dimension, I'm referring to the same kind of dimensions as when you talk about a drawing or painting being in 2D, or the 3D of real objects, that kind of dimension. And a fractal is something that isn't 2D, or 3D, or even 1D, but something that falls in between, a fractional dimension.
OK, well, I can't help myself, just a short explanation:
If I ask you, how long is the coastline of England, you could get out a map and try to measure it with a ruler. But since you can't measure all the cracks and crevices with any detail, you'll just get a rough estimate. If you go there and drive along the coast, you'll get a larger number because you'll wind around a lot, making it longer. As you get more and more accurate in measuring the length of the coastline, you keep finding it gets longer and longer, until eventually you'll say it's infinitely long. But how can it be? It's not really inifinite in length. And if I try to measure the length in 2 dimensions, you would say it doesn't have any, because a measurement of length doesn't have any "area". So fractal geometry came to the rescue. It has a "length", but not in one dimension. The length exists in a fractional dimension. And fractals were created to try to describe these fractional dimensions.
Many fractals are artificial mathematical constructs, like the Mandelbrot set, but many have been found that describe things in nature, like the structure of a snowflake, or the shape of a cloud, etc. If you try to measure the length of the edge of the Mandelbrot set, as you zoom in, it grows in length just like the example of the coastline. And that's what defines a fractal. Fractals all exist in fractional dimensions. There are all kinds of fractals, and nature is full of them, but the one thing they have in common is that they are described by fractional dimensions.
I hope that answers your questions, Undertoad.