We found that the best previously known flat map projection for the globe is the Winkel tripel used by the National Geographic Society, with an error score of 4.563. A globe of the Earth would have an error score of 0.0. The lower the score, the smaller the errors and the better the map. The Goldberg-Gott error score (sum of squares of the six normalized individual error terms) for the Mercator projection is 8.296. The object here is to find map projections that minimize the sum of the squares of the errors-a technique that dates back to the mathematician Carl Friedrich Gauss. Ignoring one of them can lead you to bad-looking maps no one would prefer. North America is lopsided to the north: Canada is bigger than it should be, and Mexico is too small. His route looks bent on a Mercator map-a flexion error.

A pilot flying a great circle route straight from New York to Tokyo passes over northern Alaska.

The Mercator map has a boundary cut error: one makes a cut of 180 degrees along the meridian of the international date line from pole to pole and unrolls the Earth’s surface, thus putting Hawaii on the far-left side of the map and Japan on the far-right side of the map creating an additional distance error in the process. Greenland appears as large as South America even though it covers only one seventh the area on the globe. It has perfect local shapes but is bad at depicting areas.

These are illustrated by the famous Mercator projection, the base template for Google maps. Previously, Goldberg and I identified six critical error types a flat map can have: local shapes, areas, distances, flexion (bending), skewness (lopsidedness) and boundary cuts. But flat maps are easy to store and manufacture and are therefore desirable. Depicting the curved surface of the Earth on a flat map has been the cartographers’ problem for centuries. I and my colleagues Dave Goldberg and Bob Vanderbei (who invented the “ Purple America” map for showing election results) have produced what we believe to be the most accurate flat map of the Earth ever made. Since becoming an emeritus professor at Princeton, I have fondly returned to some of my childhood interests. When I was 14, I made a painted globe of Mars based on a flat Mercator Mars map by the astronomer E. As a kid I was fascinated by map projections. I usually work on general relativity and cosmology.

Flat earth gleason map