Maps and Math

Gauss’s Theorema Egregium was in the news recently! The news articles didn’t quite put it that way, though. Their headlines were more like, “Boston public schools map switch aims to amend 500 years of distortion.” That’s right, they’re switching from using the Mercator map projection to the Gall-Peters projection in their classrooms.

The Gall-Peters map projection. Credit: Strebe, via Wikimedia Commons. CC BY-SA 3.0

The Theorema Egregium is the theorem that states that Gaussian curvature is an intrinsic property of a surface, not a result of how it is embedded in space. An application of the theorem is that accurate maps are impossible. The surface of the earth is positively curved, and a plane has zero curvature. So any function that maps the surface of the earth to the plane must distort something, whether it’s area, shape, or distance.

The familiar Mercator projection projects the globe onto a cylinder. It preserves shapes fairly well (circles near the poles and circles near the equator are all circular), and it is useful if you’re navigating across an ocean using a compass. But the area distortions are severe. In real life, Africa is 14 times as large as Greenland, a fact that is not clear on the Mercator map. (Check out The True Size to compare the actual sizes of different countries by dragging them around a Mercator map.) The Gall-Peters projection replacing it preserves area but distorts shape dramatically.

As a representative of the fictional Organization of Cartographers for Social Equality explains on an episode of The West Wing, the Mercator projection can make places close to the Equator, such as Africa and South America, look smaller and therefore less important than northern North America and Europe. An equal-area map makes Africa in particular seem proportionally much larger than it seems with the Mercator.

I personally find the Gall-Peters projection bracing. It looks strange and a little wrong, which helps me think about the assumptions we’ve internalized about what the world looks like. Even though I know in my head that the Mercator is inaccurate, I’m exposed to it so much that it seems like the real thing to me. But I wonder why Boston schools and writers for The West Wing chose Gall-Peters instead of a different projection that distorts shapes less. There are so many other options!

Ernie Davis, a computer science professor at NYU, wrote a guest post on Cathy O’Neil’s blog about the recent map projection discussion. He’s a member of the pro-globe camp. But it’s not always practical to have a 3-dimensional globe handy. Luckily there are several posts in the science blogosphere that give nice overviews of some of the common projections. Dave Goldberg votes for the Winkel-Tripel, which also happens to be the projection used by National Geographic, on his blog A User’s Guide to the Universe. Max Galka at Geoawesomeness votes for the Authagraph, which I had never seen before.

Mike Bostock has a great animation of dozens of the most popular map projections so you can see how they compare to one another. After you’ve found your favorite, you can see what it says about you in this xkcd comic.

Several map-related blogs have found their way into my (mostly mathematical) blog reading list. They don’t often mention math specifically, but they all combine geometry, data analysis, and data visualization with culture, history, and sociology. A few of my favorites are Musings on Maps by Daniel Brownstein, All Over the Map by Betsy Mason and Greg Miller, and Adventures in Mapping by John Nelson. I also enjoy William Rankin’s website A particular favorite is the map called Actual European Discoveries. Today these territories have a combined population larger than all of Connecticut! For dessert, I’m partial to the Twitter account @TerribleMaps, which shares helpful charts such as this map of the population per capita of countries in Europe.

Do you have a favorite map blog?

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