i spent a very enjoyable math binge day learning about waves and what i learned is that waves are extremely fucked up
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This is like the tip of the iceberg of wavy fuckedness
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For some absolute cosmic bullshit, look into: Huygen’s principle, “optical lenses as Fourier transforms” and, maybe especially, like: “fourier transform wavefunction”
It’s been a while, but yeah the universe is wild and waves are cool
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one thing i’m starting to suspect is that linear algebra is much more insane than it looks to students in an intro course but you don’t really see the insanity unless you do computations with gigantic matrices that, like, approximate differential operators
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Yes! Eigenvalues are magic. I only grokked differentiation of functions when I realized it produces the locally best fit linear transformation. And many vector spaces are way, way stranger than the R^n we tend to imagine...
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do you have an example of this?
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take the vector space of, say, continuous functions on the 2-sphere S^2. it has a special “basis” consisting of what are called spherical harmonics and the shape of spherical harmonics describes (part of) the shape of atomic orbitals:
hold on isnt the 2-sphere just a circle? wouldnt a function on it be a two dimensional function of the angle?
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1-d basis waves: guitar string harmonics etc
2-d basis waves: chladni patterns, speaker cone engineering (rabbithole warning ahead)
3-d basis waves: spherical harmonics
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and the part that still continued to bless my mind is how transforms allow you to “just figure out” which basis waves are present.
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