Goofing on Numbers: Skyscraper BMI

This is the start of Goofing on Numbers, a little semi-regular series I’d like to do here. The basic idea here is to take equations and gently misuse them to answer meaningless questions, sort of like What If? if it cared less about describing any actual physical reality. It’s an illustration of the principle of Garbage In, Garbage Out, but more importantly for my purposes, it’s tons of fun. Today I’m going to take a look at Body Mass Index, that often criticized metric of weight.

Body Mass Index, or BMI, is supposed to be a measure of how healthy someone’s weight is. Too low and they’re underweight, too high and they’re overweight. (I’m getting the specific ranges from the American Cancer Society, so they should be trustworthy.) There are a lot of criticisms of its usefulness. I won’t get into them here, since this is more for fun than entertainment, but they’re out there.

A person’s BMI is their weight in kilograms divided by their height in meters squared. The question I asked that sent me down this rabbit hole is, “What if we applied this equation to things that aren’t people?” I spent a fair amount of time idly wondering where a skyscraper would fall on the scale. My first instinct was that the mass of a skyscraper grows faster than its height, so a skyscraper would be super high on the scale. Then I thought that, since the height is squared, the denominator of this fraction grows faster than the numerator, so skyscrapers would become more “underweight” the taller they get. Skimming over the Wikipedia page for BMI, I’m beginning to suspect that my first intuition was right, but there’s no substitute for proper calculation. So, without further ado…

(Fair warning – I’m not doing super deep dives for this information, so some of it may be inaccurate. There are three different units called a ton, for instance, so it’s kind of hard to tell which is which sometimes. Sorry!)

The Empire State Building has a BMI of around 1,685. (That’s including the spire as part of the height, too, which is a little generous.) The Burj Khalifa, meanwhile, has a BMI of around 565. (Side note – in my research, I found out that there’s a restaurant near the top of the Burj Khalifa called At.mosphere. I’m not a huge fan of the name. It sounds like the name of a PS2 game about a bunch of teenagers who travel into the Internet to save the world with the help of some friendly aliens.) The Washington Monument clocks in at around 2,890, and the Eiffel Tower at around 70, which is the only one of these results that’s actually a semi-reasonable human BMI – the world’s heaviest man had a BMI of 186 at one point. A lot of these numbers are more than a little disappointing. It was too much to expect them to be within the realm of human possibility, but a girl can dream, can’t she?

This wasn’t a particularly productive exercise. That’s alright – nobody can be all business all the time. That’s why they invented the mullet, after all. Still, it was kind of fun to dig into what this statistic represents in a physical sense. (The units themselves imply something interesting – kg/m2 seems like a ratio between mass and surface area, or the density of Flatlanders.)

While we’ve had some fun here, the real moral of the story is this: if you shake equations hard enough, sometimes you can get the sentence “the Eiffel Tower is morbidly obese” to fall out, and that’s a good enough reason to mess around with math.

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