engineering, math, physics, thought experiment

The Treachery of Plumb-Lines

I’m pretty sure that’s my most pretentious article title to date, but really, the only pretentious thing about it is that it’s a Rene Magritte reference, because if you read it literally, that’s exactly what this article is about.

Imagine two skyscrapers. Both start from ordinary concrete foundations 100 meters by 100 meters, and each will be 1,000 meters high, when finished. We’ll call the first skyscraper Ruler, and the second skyscraper Plumb, for reasons I’ll explain.

Ruler is built exactly according to architectural specifications. Every corner is measured with a high-grade engineer’s square and built at precisely 90 degrees. Importantly, Ruler is constructed so that every floor is precisely 10 meters above the previous one, and every floor is 100 meters by 100 meters. This is done, of course, using a ruler. Because it’s kept so straight and square at every stage, Ruler is a very straight, square building.

Plumb, on the other hand, is kept straight and square using one of the oldest tricks in the architect’s book: the plumb-bob. True story: plumb-bobs are called that because, back in the day, they were almost always made of lead, and the Latin for lead is plumbus (or something like that; I took Latin in high school, but the teacher got deathly ill like two weeks in, so I never learned much). A well-made and well-applied plumb-bob is an excellent way to make sure something is absolutely vertical.

The builders of Plumb do use a ruler, but only to mark off the 10-meter intervals for the floors. They mark them off at the corners of the building, and they make sure the floors are perfectly horizontal using either a modified plumb-bob or a spirit level (which is largely the same instrument).

One might assume that Plumb and Ruler would turn out to be the exact same building. But anybody who’s read this blog knows that that’s the kind of sentence I use to set up a twist. Because Plumb was kept straight using plumb-bobs, and because plumb-bobs point towards the center of the Earth, and because the 100-meter difference between the east and west (or north and south walls) gives the bobs an angle difference of 0.009 degrees, Plumb is actually 11 millimeters wider at the top than at the bottom. Probably not enough to matter in architectural terms, but the difference is there.

Not only that, but Plumb’s floors aren’t flat, either, at least not geometrically flat. The Earth is a sphere, and because Plumb’s architects made its floors level with a spirit level or a plumb-bob, those floors aren’t geometrically flat: they follow the spherical gravitational equi-potential contours. Over a distance of 100 meters, the midpoint of a line across the Earth’s surface sits 0.2 millimeters above where it would were the line perfectly, geometrically straight. This difference decreases by the time you reach the 100th floor (the top floor) because the sphere in question is larger and therefore less strongly curved. But the difference only decreases by around a micron, which is going to get swamped out by even really small bumps in the concrete.

“Okay,” you might say, “so if you blindly trust a plumb-bob, your building will end up a centimeter out-of-true. What does that matter?” Well, first of all, if you came here looking for that kind of practicality, then this blog is just gonna drive you insane. Second, it doesn’t matter so much for ordinary buildings. But let’s say you’re building a 2,737-meter-long bridge (by total coincidence, the length of the Golden Gate Bridge). If you build with geometric flatness in mind, your middle pier is going to have to be 14.7 centimeters shorter than the ones at the ends. That’s almost the length of my foot, and I’ve got big feet. It’s not a big enough difference that you couldn’t, say, fill it in with concrete or something, but it’d certainly be enough that you’d have to adjust where your bolt-holes were drilled.

What’s the moral of this story? It’s an old moral that probably seems fairly ridiculous, but is nonetheless true: we live on the surface of a sphere. And, when it comes down to it, that’s just kinda fun to think about.

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