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.

Relat­ed Con­tent:

The His­to­ry of Car­tog­ra­phy, the “Most Ambi­tious Overview of Map Mak­ing Ever,” Now Free Online

New York Pub­lic Library Puts 20,000 Hi-Res Maps Online & Makes Them Free to Down­load and Use

Browse & Down­load 1,198 Free High Res­o­lu­tion Maps of U.S. Nation­al Parks

Down­load 67,000 His­toric Maps (in High Res­o­lu­tion) from the Won­der­ful David Rum­sey Map Col­lec­tion

Free: Nation­al Geo­graph­ic Lets You Down­load Thou­sands of Maps from the Unit­ed States Geo­log­i­cal Sur­vey

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities and cul­ture. He’s at work on a book about Los Ange­les, A Los Ange­les Primer, the video series The City in Cin­e­ma, the crowd­fund­ed jour­nal­ism project Where Is the City of the Future?, and the Los Ange­les Review of Books’ Korea Blog. Fol­low him on Twit­ter at @colinmarshall or on Face­book.

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Comments (3)
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  • Bane says:

    Its inter­est­ing to learn, but sad to know that the world does­n’t teach maps cor­rect­ly enough at school that any­one would need this to be re-taught to them via the inter­net.

    I was taught that the only real cor­rect view, was when the place you were look­ing at was at the cen­tre, and every­thing around it was dis­tort­ed. Hav­ing Amer­i­cans look­ing at the map from the African/British point of ref­er­ence is skewed at best.

  • Kevin Johnson says:

    Anoth­er inter­est­ing writer on a slight­ly relat­ed sub­ject, deter­min­ing lon­gi­tude, is Umber­to Eco, who wrote _The Island of the Day Before_.

  • Chris Merle says:

    I can’t believe they left out R. Buck­min­ster Fuller’s Dymax­ion Map which is a very accu­rate project of the Earth on a flat sur­face. Basi­cal­ly it’s a flat­tened icosa­he­dron. Wikipedia gives a good overview (https://en.wikipedia.org/wiki/Dymaxion_map).

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