I trust you’re having a relaxing Sunday? Mathematical physics can be relaxing too, especially when you just look at it. We’re just going to look at this one. In fact, this is a literal mathematical instantiation of Sunday relaxation.

If you fix a wire or a rope at two points and let it hang naturally, it forms a shape called a catenary.

It looks a lot like a parabola, and it turns out that in fact as long as the suspension points aren’t too close together compared to the length of the rope, it’s a very good approximation. For this particular graph the error from a purely parabolic approximation would be nowhere greater than 4%.

What’s physically interesting about this curve is that the sum total of the potential energy of each little part of this rope is a minimum. There’s no possible way to arrange the curve to have a lower potential energy. Nature does it automatically.