I usually work on general relativity and cosmology. I have always loved geometrical things. As a kid I was fascinated by map projections. When I was 14, I made a painted globe of Mars based on a flat Mercator Mars map by the astronomer E. M. Antoniadi. Since becoming an emeritus professor at Princeton, I have fondly returned to some of my childhood interests. 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.

Flat Earth - Great Circle Routes on Gleason Map – GeoGebra Flat Earth - Great Circle Routes on Gleason Map – GeoGebra

The Winkle tripel is a map to hang on your wall. Ours is a more accurate one you can hold in your hand. Steffen also said that individuals can perform a similar experiment using geometry and Polaris, or the north star, as a function of latitude. The relationship between the coordinates ( θ, ρ) of the point on the globe, and its latitude and longitude coordinates ( φ, λ) is given by the equations: Goldhaber-Gordon added that this experiment would show that “locations close to the south pole appear far apart on the map,” although that is not the reality. A disadvantage of the new map is that you can’t see all of the Earth’s surface at once, but remember this is true for the globe as well. Our map is actually more like the globe in this respect than other flat maps. To see all of the globe, you have to rotate it; to see all of the new map, you simply have to flip it over, as you can see belowOne 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. Azimuthal equidistant map projection An azimuthal equidistant projection about the North Pole extending all the way to the South Pole. An azimuthal equidistant projection about the South Pole extending all the way to the North Pole. Emblem of the United Nations containing a polar azimuthal equidistant projection. Some social media users are saying that Alexander Gleason’s 19th Century “New Standard Map of the World” is proof that the earth is flat and that Antarctica is not a continent but an ice ring that circles the earth’s edges. They are wrong. The earth is not flat. The map has been misinterpreted. In May 1893, Gleason patented the invention of the “New Standard Map of the World”, a projection of the earth centered on the north pole, which can be found ( here). Gleason might have claimed that the map is the “flat-Earth map” but his explanation written in his patent for the map is contradictory:

Map Gleason 1892 World Time Calculator Flat Earth Large Wall Map Gleason 1892 World Time Calculator Flat Earth Large Wall

This double-sided map has a Goldberg-Gott error score of only 0.881 versus 4.563 for the Winkel tripel. It beats the Winkel tripel in each of the six error terms! 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.

SNYDER, John P. (1997). Flattening the earth: two thousand years of map projections. University of Chicago Press. ISBN 0-226-76747-7. , p.29 With the circumference of the Earth being approximately 40,000km (24,855mi), the maximum distance that can be displayed on an azimuthal equidistant projection map is half the circumference, or about 20,000km (12,427mi). For distances less than 10,000km (6,214mi) distortions are minimal. For distances 10,000–15,000km (6,214–9,321mi) the distortions are moderate. Distances greater than 15,000km (9,321mi) are severely distorted.