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I would add to your list of reasons why Fourier Transforms are awesome.

By complete coincidence, Bragg's law, used to do everything from X-Ray Diffraction to particle scattering, just happens to be a fourier transform. Every time we bombard a tiny thing with light or radiation in order to understand the structure, what we literally get out of it is emission dots that correspond to the periodicity of the lattice -- literally the 2D fourier transform of the scattering cross section. When I heard that in Quantum III, it blew my mind. It's straight out of quantum scattering theory.




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