Every flat map of Earth must lie about something
Map projections transform a globe's curved surface onto a plane. Gauss proved a sphere cannot be mapped without distortion. Mercator preserves angles but swells polar lands; equal-area projections keep country sizes honest but bend shapes. Cartographers pick the lie that suits the map's purpose.
In cartography, a map projection transforms coordinates from a curved globe onto a plane. Projection is essential to any two-dimensional map. All such transformations distort something: conformal maps preserve angles but not areas; equal-area maps preserve sizes but warp bearings. There is no limit to possible projections, and each serves a different purpose.
Carl Friedrich Gauss's Theorema Egregium established that curved surfaces like spheres resist faithful flattening onto a plane—the same holds for ellipsoids and geoids. Nicolas Tissot's indicatrix visualizes local distortion as ellipses at each point. The Goldberg-Gott indicatrix adds flexion and skewness. Dynamic tools drag coastlines across projections to show size changes.
Creating a projection involves two steps: choosing an Earth model (sphere or ellipsoid, always a simplification), then converting latitude and longitude to plane coordinates. Cylinders, cones, and planes are developable surfaces that unroll without further tearing. Spheres and ellipsoids are not developable—you cannot flatten an orange peel intact.
The Mercator projection in normal aspect keeps scale depending only on latitude. Equal-area Sinusoidal and Gall–Peters projections show correct relative country sizes. The National Geographic Society favors compromises like Robinson and Winkel tripel. Large-scale national maps must match datum to projection; world maps tolerate tiny coordinate differences between datums. Only a globe keeps one scale everywhere in every direction; no flat map manages that for even the smallest patch. The Mercator itself drew criticism throughout the twentieth century for inflating lands far from the equator, and in the first half of that century a popular way to reveal distortion was to redraw a human head on different grids.
Source: Map projection
