Fun exercise for the reader: if you have that uniform hanging rope support a uniform flat suspension bridge (via a bunch of closely spaced vertical ropes), and the bridge is much, much heavier than the ropes, then you get a parabola instead of a catenary. Wikipedia gives a derivation involving differential equations, but it glosses over the actual fundamental difference between these situations. But you can explain what’s going on with just trigonometry and no calculus, let alone differential equations: consider how much weight a small section of chain that isn’t right in the middle is supporting. You’ll end up with a drawing involving a right triangle and some numbers associated with the sides, and those numbers will line up differently with the opposite, adjacent and hypotenuses in the two cases.

So your off-topic rambling isn’t off-topic at all :)