A globe is curved; a screen is not. Every world map ever drawn has to break something — area, shape, distance or direction — to make that transfer. Below, the same live NASA imagery rendered two different ways, then the ten great projections and exactly what each one sacrifices.
Identical VIIRS true-color data. Watch the poles stretch in one and settle in the other.
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What each projection preserves — and what it destroys to do it.
A sphere has intrinsic curvature; a flat plane has none. Gauss proved in 1827 — his Theorema Egregium, the "remarkable theorem" — that you cannot flatten a curved surface onto a plane without stretching it. Peel an orange and try to press the skin flat: it tears. That tear is every world map ever made.
So cartographers choose what to save. Conformal projections (Mercator) preserve local shape and angle, which is why they won the age of sail — and why they wildly inflate the poles. Equal-area projections (Gall-Peters, Cylindrical Equal Area) preserve true size, and pay for it by squashing shape. Compromise projections (Robinson, Winkel Tripel) distort everything slightly rather than anything badly, which is why they look "right" to us.
The politics follow the mathematics. A Mercator classroom map makes Europe and North America look far larger than the equatorial world; that is the argument Arno Peters made in the 1970s when he pushed his equal-area map as a corrective. Neither map lies on purpose — each simply chose a different thing to protect.