School geometry gives us a very simple and practical rule: parallel lines never meet and triangles add up to 180°. Handy, and almost true. The “almost” is Euclid’s small mistake. He treated physical space as perfectly flat. That works on a desk, a street, even a bridge. But stretch those lines across oceans, or send timing signals between satellites and phones, and the world’s not flat enough.
On a sphere the rules bend. “Straight lines” become great circles, like the Equator or a meridian, and any triangle you draw, say, two meridians meeting at the poles plus the Equator, has angles summing to more than 180°. That’s why long-haul flights arc over the map: they’re following geodesics on a curved Earth. Satellite navigation has to speak this spherical language (more precisely, an ellipsoidal one), or positions drift.
Einstein pushed the idea further: space and time themselves are curved by mass and motion. GPS only works because its engineers correct for this non-Euclidean reality. A satellite’s clock ticks differently from one on Earth, faster due to weaker gravity, slower because it’s moving, with a net offset of about +38 microseconds per day. Leave that “tiny” mismatch unpatched and location errors balloon by kilometers. So GPS (and Galileo/BeiDou) pre-bias satellite clocks and apply relativistic formulas continuously as signals race along bent paths in curved spacetime.
Still, Euclid isn’t obsolete, he’s still used everyday. For building a house, laying a football pitch, or sketching a neighborhood map, flat-space geometry is perfect: simple, fast, and accurate enough. The bigger lesson is methodological: useful models are often deliberate reductions. They trade universal truth for tractable insight, then flag when to switch gears.
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