mathematics is nothing but the search for structures. beautiful structures. i myself tend to think i understood something if i manage to have a picture of it in my head, some geometric interpretation. sometimes it’s more easy to understand something if one can somehow draw it, by hand, or using a computer, which can sometimes give insights which would not be possible without it; the best example maybe is the beautiful mandelbrot set. most visualizations are by no means as impressive as the mandelbrot set, unless the viewer is acquainted with the underlying mathematics. like the following one, produced by the programs i wrote yesterdays and today, whose beauty probably won’t unfold if you don’t know what it depicts:
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