## Catenary Inversion: Curves of Sagrada Familia

Sagrada Familia is stunning and beautiful. If you ever go visit, don’t miss out on the bottom level: there are exhibitions about the constructions and history of this masterpiece by Gaudi. When I visited a while ago, I was surprised to find a model that explained how the curves of the arches of the church were designed, and it was really cool.

To start out, imagine you’re building the roof of a house. Usually they are like this: /\. Modern houses also look like this: Π. The point is, a flat roof is hard to support, so the older houses are all angled. If you hold a dumbbell horizontally to your side, you’ll feel tired a lot faster than holding it angled upwards. This is because materials in general are a lot better at handling compression than bending forces. By holding your arm at an angle, you are supporting part of the weight by compressing your arm along its own direction, reducing the amount of force perpendicular to that direction. Back to the roof: a flat one is fine if made by concrete and steel, but if we use a long piece of wood, maybe not.

Anyway, using the same materials, an arch shaped building will last much longer than a flat topped one, simply because the bricks are subject to bending forces to a less extent. The problem then becomes: how can we find the shape that minimizes bending force at every point on the arch (to zero, actually)?

If you remember high school physics, we can dive into it. Say we draw an arch like this: ∩, and we pick any brick on the arch (say on the left half). Let’s pretend this is the ideal curve, such that there is no bending force anywhere. This little brick you picked is going to have a tiny bit of mass, and the slope of the arch changes a little bit before and after it. Then we have three forces acting on this dot: gravity which points down, force from the left brick supporting it, and force to the right brick. The latter two forces have slightly different slopes, and the three add up to 0 (otherwise the arch will collapse. Note that we don’t have forces perpendicular to the arch between bricks, which is the whole point.) Oh no, we have a differential equation! It’s been 2 days since I took that exam, I have forgotten everything! What do I do?

The Catenary Curve