Quad-Tree

I've been searching around for the logic behind QuadTrees, and trying to implement my own QuadTree algorithm.  What's the basic logic behind QuadTrees?

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You will call anything you don't understand pseudoscience. You do realize that you could limit your sets to finite and computable objects right? Even if a fractal could be described by a non-halting (by definition infinite) process, you only need to compute a part of it for most practical uses.