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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I think I know what you are talking about here, but tweets won't do it. What you describe as translations and rotations may be something else: equivalences required by logic, so nothing is being "preserved" per se. Happy to do a video about it if you want to dig in. Love geometry
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This is related to one of Euclid's "slips" from the axioms in the elements. He used a superimposing argument to prove a proposition, backing it up only with Axiom 4 "Things which coincide with one another are equal to one another".
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I believe the notion of "triangle congruence" was invented by the early 20-th century mathematicians who realized this, basically to fix this "mistake". It is a shame though, since triangle congruence clearly doesn't capture everything you could do with "superimposing arguments".
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Any two polygons of the same area can be subdivided into matching triangles like a tangram puzzle, and triangles can be compared for congruency by line segment length. It's maybe not obvious that area of this sort is well defined, but it is at least easy to compare for equality.
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yeah that's very clean and classical but it's famously known not to generalize to volume:
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I love this!
(As you know of c., non euclidean geometry was invented because people were like, "you keep TELLING me these lines must eventually intersect - but, like, dude, can you PROVE it?
Maybe there's a new geometry as well, in which "area" is not conserved...
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You may be ahead of me on this but isn’t that part of the definition of space? You have a metric that sets infinitesimal areas to integrate over your shape, and a connection that lets you slide around the points/lines involved. In flat space, area and shape play nicely.
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that we live in something approximating flat space is an *observation*. It could have been otherwise
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