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Voilá, the Mandelbrot set ... or rather, a very small part of it.
I won't go into too much theory here, but it's a fractal, ie an infinitely detailed structure created by a rather simple mathematical constraint over complex numbers. The Mandelbrot set is probably the best known fractal, and good friend google will find you plenty of resources.
Basically, the issue has been beaten to death already. So what's this for, I hear you asking ...

Recently a friend complained about the lack of numerical precision in commonly available Mandelbrot renderers, making them useless for what he appropriately calls 'deep zooms'.
So I picked up the complaint and created a little app that has grown over the past weeks into something really nice, I hope.
In case you're still reading, the above image was produced with it, with set coordinates close to -2/0 and a radius of about 3.0e-36. That radius number is so small, without scientific notation it would take me half a line of text just to write all the leading zeroes. Anyway, all other applications I've tried start breaking at radii around 1.0e-18. That's a lot less zeroes ;)
So this is the real purpose, it's extremely precise. This is accomplished by virtue of a custom 224 bit floating point format. The others seem to be restricted to the 64 bit floating point formats that are native to x86 processors. Otherwise, it packs some cool usability features too. Both keyboard and mouse navigation are supported, along with a web browser-like history and saving and restoring of 'cool' locations. Selectable palettes and gamma corrected antialiasing round the package. Exact usage details can be found in the enclosed readme. Just give it a try, it won't hurt you.

With all of this behind us, you may now finally grab it here. As this is free software under LGPL terms, full source code is included. Feel free to redistribute.