still don't think i understand the pythagorean theorem, all things considered
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it would take awhile to elaborate more fully on what i mean but here's a chunk of it: all the visual / geometric proofs take for granted that whatever "area" means it's something that's preserved under translations and rotations. it's unclear exactly what is needed for this
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there's a book called "the mathematical mechanic" that provides "physical proofs" of the pythagorean theorem. they all seem to be saying the same thing about translation- and rotation-invariant physics but i don't quite get what that thing is
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here's the first proof, that derives the pythagorean theorem from the impossibility of perpetual motion (and, implicitly, that the concept of torque is meaningful and behaves nicely wrt rotations). i cannot claim to really understand what is going on here
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This is actually such a beautiful proof, I don't think I've seen it displayed this way before
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it's very tantalizing! something is obviously going on but i don't feel like i have a grasp on it quite yet
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A kinematic proof of the pythagorean theorem is a kind of "backing into" — physics are a more complicated set of rules than only geometry, so it will only make sense if you understand both geometry and rigorous physics. Again happy to dig into all of that if you're curious.
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I can't explicitly explain what I felt I understood but when I saw Figures 2.1 & 2.2 my jaw dropped and I made a "ungh!" sound of realization. I think some part of me gets the proof.
(I studied torque in engineering, so I have some intuitions for it.)
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Reading the actual description, yeah, the math checks out too. Not sure I've fully hooked up the math to the intuition moment (legit no pun intended!) but yeah, this is really cool.
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Fascinating. Some early 2000s games had a physics engine that did not have laws like this and were notorious for speed runners accelerating indefinitely by jumping and rotating mid-air
Here's one proof of this "strafe jump" effect in a Star Wars game
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Where's the rest of it? You can derive the motion of the planets from straight geometry.
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What shape tank would rotate? Does the answer imply that the theorem applies to all shapes? Or maybe that all shapes are just collections or triangles?