(Geographic) Overcompensation and Projection

Buzzfeed posted something a couple weeks ago that’s been in my head ever since: maps of the US with the outline of other countries superimposed. Sometimes, the result is uninstructive:

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Japan is distinctly smaller than the US. I bet you knew that.

Sometimes, however, It really took me by surprise. For example, I tend to think that Russia dwarfs the US in size. And it is considerably larger. But whereas I might have guessed it was at least three times larger, in reality it’s not even two:

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Similarly, I think of Madagascar as a smallish island off the coast of Africa (where annoying, animated animals inexplicably get an endless stream of sequels to their mediocre movie as a result of one catchy dance track.)

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But Madagascar is comparable to most of the East Coast. It’s larger than California and fairly close to the size of Texas.

Why was I so wrong about these? Two reasons. First, context matters. If you live in America, you frequently see maps of just the US. However, you’re unlikely to see a Madagascar map without the rest of Africa. Since Africa is huge (three times the size of the whole US) then of course you end up thinking Madagascar is small.

The other problem is bigger and more widely discussed. Literally no map you’ve ever seen is accurate, unless it’s actually a globe. You cannot change a globe into a map without ruining its accuracy in at least one way and probably more. You can picture this if you’ve ever peeled an orange: There’s definitely no way you can flatten out the skin into a nice, connected rectangle. At best, it might look something like this:

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Stolen shamelessly from Nathan Belz’s compelling and much-too-short blog at transpographics.blogspot.com. Thanks Nathan; this is fantastic.

So how do you make this a rectangle? An obvious way (called a “Mercator projection*”) is to unwrap the map like this…

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…and then basically stretch the pointy parts until they meet up. As you can see, this requires a lot of stretching near the poles, and not much near the equator. Famously, this makes Greenland look huge–as large as Africa on some maps! And I think it’s the key reason I overestimate the size of northerly Russia.

Incidentally, the projection problem also manifests itself in another way that always bothered me as a little kid growing up in Colorado. Not just areas and distances, but also angles can be distorted. As a result, I used to be really confused about whether my state was a rectangle or not.

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But this one makes it look a little saggy, like an old futon.

Which is it, Colorado? Pick a side!

Turns out, Colorado is not a rectangle. It’s borders were meant to follow lines of latitude and longitude, so the left and right sides bend towards each other slightly towards the top. But the other map is “wrong” too– the top and bottom sides do not curve down slightly. They follow lines of latitude, which are all parallel to the equator. **

The bottom line is, no two-dimensional map will ever fully and accurately capture a three-dimensional sphere. In case you needed another way to use the globe on your desk to make you seem superior.

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*Actually, what I’ve just described is a whole category of map-stretching strategies called “cylindrical projections.” It’s not even exactly that, to be honest, but it’s very close and it’s a good way to picture it. The exact nature of how you cut the pointy slices and how much you stretch in between them determines whether it’s a Mercator projection or some other kind of cylindrical projection. But the Mercator is by far the most common and it’s the term you’re most likely to see elsewhere.

You can read more about the problem of map distortions and the various techniques for minimizing them here. It’s slightly technical but it also contains some ugly-but-instructive pictures that nicely demonstrate the kind of skewing that can occur. Basically, they imagine drawing a bunch of identical circles on the globe before doing the projection that turns it into a map, so by looking at how the circles, once uniform, have changed relative to one another, you can see how the map features are skewed. The “impress your friends” term for these circles is “Tissot’s Indicatricies.”

** To be fair to the mapmakers, there’s a justification for each choice above, and it’s my decision to mix and match criteria which makes them seem more weird than helpful. For example, the “rectangular” Colorado reflects the fact that the borders really do point true north/south and east/west. And the top and bottom borders, while indeed parallel to the equator, are not really flat– they curve, like everything, around the surface of the earth. So which one represents it better? It depends what you want to use the map for. All we can say for sure is that the “real” Colorado is neither of the two shapes, because it is not a two dimensional shape at all. It is a curve, unflattenable slice of orange peel that looks kind of like a rectangle, but really more of a trapezoid, when viewed from certain angles.

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About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

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