How is the most accurate flat map of the Earth

“Normally, I work in general relativity and cosmology” – explains J. Richard Gott, Professor Emeritus of Astrophysical Sciences at Princeton University and one of the people responsible for this incredible breakthrough in world cartography. I have always liked geometric things. When I was a child, I was fascinated by map projections. When I was 14 years old, I made a terrestrial globe painted on the basis of a flat map of Mercator Mars by the astronomer EM Antoniadi. Since becoming a professor emeritus, I have fondly returned to some of my childhood interests. “

As recounted in a recent report published in American scientist, with his colleagues Dave Goldberg and Bob Vanderbei (who invented the “America purple” map to show election results) they produced what they believe to be the most accurate flat map of Earth ever. Representing the curved surface of the Earth on a flat map has been the problem of cartographers for centuries. Neither can be perfect, but they are easy to store and manufacture, and therefore desirable.

Previously, Goldberg and Gott identified six types of critical errors that a planar map can have: local shapes, areas, distances, bending (curvature), tilt (deflection) and boundary cuts. These are illustrated by the famous Mercator projection, the basic model of Google Maps. It has perfect local shapes, but it’s bad at representing areas. Greenland looks as big as South America even though it only covers a seventh of the world’s land area.

“You can’t do everything perfectly,” Gott explains. The Mercator map has a boundary cut error: we make a 180 degree cut along the meridian of the international date line from pole to pole and unroll the surface of the Earth, thus placing Hawaii on the left side of the map and Japan on the far right. side of the card creating an additional distance error in the process“. A pilot flying a circular route directly from New York to Tokyo passes over northern Alaska. Your route appears to be folded on a Mercator map – a curvature error. North America is out of balance in the north: Canada is bigger than it should be and Mexico is too small.

The plain of the circle

The goal they set for themselves was find map projections that minimize the sum of squares of errors, a technique dating back to mathematician Carl Friedrich Gauss. The Goldberg-Gott error score (sum of the squares of the six individual normalized error terms) for the Mercator projection is 8,296. The lower the score, the fewer errors and the better the map. . A globe would have an error score of 0.0.

“We found that the most well-known planar map projection in the world to date is the Winkel tripel used by the National Geographic Society, with an error score of 4,563,” continues the professor. It has straight pole lines at the top and bottom with curved left and right margins that mark its 180 degree border in the middle of the Pacific. “

They seem to have reached the limit of the Winkel tripel upgrade. When this happens in science, it often takes groundbreaking and innovative thinking to make radical progress. Richard Feynman once said that in physics, when we’re stuck, when all the old ways don’t work, the new trick, the new way that is going to work, will be very different from anything we’ve seen before.

“The idea for the new map projection came from a recent article I wrote,” says Gott, “Envelope Polyhedra,” which introduces a new class of figures in which polygons can appear one after another. I realized I could make an attached circular map: like an old record. One side of the map shows the northern hemisphere, the other side shows the southern hemisphere, with the equator circling around the edge. There are no contour cuts and the correct topology of a sphere. You simply “crush” the flat sphere. “

With that in mind, they had to find the best formula to plot the characteristics of each side, one that will minimize the Goldberg-Gott error score. The answer is one in which the North Pole appears in the center of the north side of the disk, with lines of longitude also extending where the scale on each line of longitude is uniform, and the same for the south side. Distances between cities are measured simply by stretching a chain between them; if they are in opposite hemispheres, the chord extends across the equator to the edge of the map.

A card that looks almost perfect

One downside of the new map is that you can’t see the entire surface of the Earth at once, but this is also true for the world. The new map looks more like the globe in this respect than the other planes. To see the entire globe, you have to rotate it.

This double-sided card has a Goldberg-Gott error score of just 0.881 versus 4.563 for the Winkel tripel. Surpass it on each of the six error terms. It has a zero limit cutoff error since the continents and oceans are continuous on the circular edge. It has a remarkable property that no one-sided planar map has: the distance errors between pairs of points (like cities) are delimited, with a difference of only plus or minus 22.2%. In the tripel projections of Mercator and Winkel, the distance errors skyrocket as one approaches the poles and cuts to the limits.

“Our cards can be cut out or inserted in loose sheets in a magazine. They can be cardboard or plastic. A thin box could contain double-sided flat maps of all major solar system objects, or a stack of Earth maps with physical and political data. The Winkle tripel is a map to hang on the wall. Ours is more precise than you can hold in your hand, ”enthuses Gott.

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