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This is a great video, thanks. The mentioned catenary shape is a cosh (Cosinus Hyperbolicus, i.e. 1/2(exp(x) + exp(-x)) or cos(ix)) and decidedly not a parabola.

Wikipedia contains a derivation: https://en.m.wikipedia.org/wiki/Catenary (the one on the German Wikipedia is a bit more explicit)

https://de.m.wikipedia.org/wiki/H%C3%A4ngemodell https://de.m.wikipedia.org/wiki/Kettenlinie_(Mathematik)

NB, the line taken by the cables of a suspension bridge with a deck of uniform weight is a parabola. I think this is what causes the popular confusion.

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