Man, that's some nice resolution 
Every time I've implemented it I run into rounding errors after several orders of magnitude of zoom... makes the image look just wrong.
They must have some clever floating point algorithms hidden inside that program 
I thought Wikipedia would explain to me what you are on about, all I got for my trouble was a headache.
...and it helps expose the maths geeks
Mandelbrot setFrom Wikipedia, the free encyclopediaJump to: navigation, search
"Initial image of a Mandelbrot set zoom sequence with a continuously coloured environmentThe Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape. The set is closely related to the Julia set (which generates similarly complex shapes), and is named after the mathematician Benoît Mandelbrot, who studied and popularized it.
More technically, the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded.[1] That is, a complex number, c, is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets.
For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = i (where i is defined as i2 = −1) gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i, ..., which is bounded and so i belongs to the Mandelbrot set.
Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the whole set, and not just to its parts.
The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization."
A fractal in layman's terms is an equation that describes a never ending hole in reality or a pattern that mostly replicates itself when iterated.
What you're seeing here is a group of people who have tried to see how far the whole goes.
Imagine you're looking at the known universe... the milkyway is just a tiny speck, you zoom in until you can see and electron... they may actually be zooming in further than that.
The pattern and colours are plotting lines and colours with numbers.
FYI: The wiki description is much better.
While fractals can make pretty pictures, they also started a new branch of mathematics which has helped us to understand the crazy world we live in.
It all began when some guy (think someone like Laurie but way geekier and not nearly as cool) was trying to model the weather. He had logged into his trusty campus computer, and had plugged in some equations for a really simple world. This planet was completely smooth like a big billiard ball. All it had were variables describing wind, sun and ocean.
Also being before the advent of computer screens the only output device he had was a dot matrix printer. This was quite limited in what it could output, so he just output the equivalent of the seabreeze graph on the paper. He'd start it going, and marvel at how the wind would go up, down, swing around, without a definite repeating pattern emerging. Other people at the campus would come around and gaze in awe at the strange patterns emerging from the printer. I guess people were easily amused in those days.
Anyway one day the computer stopped for some reason, so he plugged in the results that were printed out a page or two from the end of the printout and started the process going again. To his utter amazement the pattern started out the same as how the process had gone previously, but then rapidly diverged into a completely new system. This was the eureka moment, when the first glimmerings of chaos theory was born.
The mandelbrot set was invented by some french dude without even a computer to plug equations into. He just thought really hard about stuff, and invented the mandelbrot set. He even managed to draw some rough diagrams of what it might look like. That's pretty impressive...
If you're wondering how to make it yourself, firstly make a plane of numbers, with real numbers (-3 to 3 works well) going sideways, and imaginary numbers (again, -3i to 3i works well) going vertically.
Now pick a point anywhere on this plane. It will consist of two coordinates, and we'll call it c for the moment.
Now you have your starting point, repeat this equation 20 times:
z = z squared + c
(where z starts at 0 + 0i ).
If, after 20 iterations the number you're left with is a long way from the origin then paint that spot white, if it's still close to (0, 0i) then paint it black.
A really good book to sink your teeth into is called "Chaos", by James Gleick. Highly recommended for the geekier amongst us 
i guess u either get it or u don't. kind of like maths at school.
teach wats da point of dis?
Butthead: huha boobs
Beavis: quit it butthead
I need tp for my bung whole
Reality check - I think I might be an uneducated geek as I'm actually enjoying this discussion and loved the fractal zoom video but am still trying to get my head around the whole thing.
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