The problem with flat maps
Every world map is, in a sense, a lie, and it cannot be otherwise. The Earth is very nearly a sphere, and a sphere cannot be spread out flat without tearing, stretching or squeezing some part of it. A mapmaker who wants a neat rectangle of paper or screen must therefore accept a distortion somewhere; the only real choice is which one. This is not a failing of skill or care. It is a mathematical certainty, and every flat map ever drawn is simply a particular answer to the same impossible problem.
The impossibility has a precise basis in mathematics. A sphere and a flat plane have different curvatures, and in the nineteenth century Carl Friedrich Gauss proved that no method exists for turning one into the other while keeping every distance unchanged. Something must give. A projection, as such a method is called, can preserve area, or shape, or direction, or the scale along certain lines, but it can never preserve all of them at once. To gain one property is to give up another, and the mapmaker’s real task is the choosing of a loss.
The most famous answer is the projection published by Gerardus Mercator in 1569, and it was built for a single purpose: navigation. On a Mercator map a course of constant compass bearing appears as a straight line, so a sailor could rule a line between two ports and read off the heading to steer. For crossing an open ocean by compass this was invaluable, and for four centuries it made the projection the natural choice for sea charts. Within the job it was designed to do, it is close to perfect.
That usefulness comes at a steep price in area. To keep directions true, Mercator stretches the map more and more towards the poles, so that lands far from the equator swell well beyond their true size. Greenland, on such a map, looks about as large as Africa, although Africa is in reality some fourteen times larger. Because the projection was for generations the one hung on classroom walls, critics have argued that it quietly taught a distorted picture of the world, inflating the northern latitudes where the wealthier nations happen to lie.
The obvious remedy is a projection that keeps areas honest, and several exist. Equal-area maps such as the Mollweide, or the Gall–Peters that was once promoted as a corrective to Mercator, show every country at its true relative size. But the trade-off returns in a new form: to fix the areas, these maps must distort shapes, so continents appear stretched or squashed in ways that look wrong to an eye trained on Mercator. Neither map is more correct than the other. Each simply keeps the one promise it has chosen to make.
Between the extremes sit the compromise projections, which try to be moderately wrong about everything rather than exactly right about one thing. The Robinson and the Winkel Tripel spread the unavoidable error across the whole map, so that no single property is perfect but none is badly betrayed. The result pleases the eye, which is why atlases tend to favour them for general world maps. The deeper lesson is that there is no true map, only a map fit for a purpose: one projection for navigation, another for comparing the sizes of nations, a third simply for a pleasant overview.
The story has a modern twist. When online maps arrived, their designers revived Mercator, not out of nostalgia but for a technical convenience: the projection turns the world into a neat square grid that can be sliced into tiles and zoomed smoothly at any scale, and it keeps small shapes locally accurate, which matters when you are looking at a single street. So the very projection once criticised for misrepresenting the globe is now the one that billions of people see every day on their screens. The old compromise has not been solved. It has simply moved to a new home.