Tuesday, December 6, 2016

Why Making Accurate World Maps Is Mathematically Impossible

Jorge Luis Borges once wrote of an empire wherein “the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province.” Still unsatisfied, “the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it.” But posterity, when they lost their ancestors’ obsession for cartography, judged “that vast Map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters.” With that enormous map, in all its singular accuracy, cast out, smaller, imperfect ones presumably won the day again.

With that well-known story “On Exactitude in Science,” Borges illustrated the idea that all maps are wrong by imagining the preposterousness of a truly correct one. The Vox video “Why All World Maps Are Wrong” covers some of the same territory, as it were, first illustrating that idea by slitting open an inflatable globe and trying, futilely, to get the resulting plastic mess to lie flat.

“That right there is the eternal dilemma of mapmakers,” says the host in voiceover as the struggle continues onscreen. “The surface of a sphere cannot be represented as a plane without some form of distortion.” As a result, all of humanity’s paper maps of the world–which in the task of turning the surface of a sphere into a flat plane need to use a technique called “projection”–distort geographical reality by definition.

The Mercator projection has, since its invention by sixteenth-century Flemish cartographer Gerardus Mercator, produced the most widely-seen world maps. (If you grew up in America, you almost certainly spent a lot of time staring at Mercator maps in the classroom.) But we hardly live under the limitations of his day, nor those of the 1940s when Borges imagined his land-sized map. In our 21st century, the satellite-based Global Positioning System has “wiped out the need for paper maps as a means of navigating both the sea and the sky,” but even so, “most web mapping tools, like Google Maps, use the Mercator” due to its “ability to preserve shape and angles,” which “makes close-up views of cities more accurate.”

On the scale of a City, in more Borgesian words — and probably on the scale of a Province and even the Empire — Mercator projection still works just fine. “But the fact remains that there’s no right projection. Cartographers and mathematicians have created a huge library of available projections, each with a new perspective on the planet, and each useful for a different task.” You can compare and contrast a few of them for yourself here, or take a closer look of some of the Mercator projection’s size distortions (making Greenland, for example, look as big as the whole of Africa) here. These challenges and others have kept the Disciplines of Geography, unlike in Borges’ world, busy even today.


  1. Unless your campaign world is flat

  2. That flat world is, of course, supported on the backs of four elephants, themselves resting on the back of a turtle.