Projecting the surface of a globe on to a flat surface always introduces distortions. Richard Gott, Dave Goldberg, and Bob Vanderbei claim to have created created a projection that minimizes the errors more than any projection before.
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. These are illustrated by the famous Mercator projection, the base template for Google maps. It has perfect local shapes but is bad at depicting areas. Greenland appears as large as South America even though it covers only one seventh the area on the globe.
One can’t make everything perfect. 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. A pilot flying a great circle route straight from New York to Tokyo passes over northern Alaska. His route looks bent on a Mercator map—a flexion error. North America is lopsided to the north: Canada is bigger than it should be, and Mexico is too small. All these errors are important. Ignoring one of them can lead you to bad-looking maps no one would prefer.
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. The Goldberg-Gott error score (sum of squares of the six normalized individual error terms) for the Mercator projection is 8.296. The lower the score, the smaller the errors and the better the map. A globe of the Earth would have an error score of 0.0. 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. It has straight pole lines top and bottom with bulging left and right margins marking its 180 degree boundary cut in the middle of the Pacific.
These three people have created a projection that has a very small Goldberg-Gott error score of just 0.881.
It has zero boundary cut error since continents and oceans are continuous over the circular edge. It has a remarkable property no single-sided flat map possesses: distance errors between pairs of points (such as cities) are bounded, being off by only at most plus or minus 22.2 percent. In the Mercator and Winkel tripel projections, distance errors blow up as one approaches the poles and boundary cuts.
You can see their map by following the link.