On a Sphere, a Triangle Can Have Three Right Angles

Stand at the North Pole, walk straight south to the equator, turn left ninety degrees, walk a quarter of the way around, turn left again and walk back to the pole. You've walked a triangle with three straight sides and three right angles: 270 degrees, not 180. The 180° rule rests on Euclid's parallel assumption, which nobody could prove because it isn't always true. On a sphere the straightest paths are great circles, and any two of them cross twice, so there are no parallel lines. A triangle's angles add up to more than 180°, and the extra amount (in radians) is exactly its area divided by the square of the radius: our polar triangle covers 1/8 of 4πr², which gives π/2 = 90° of excess. Small triangles look flat, circles come out shorter (the equator is only 4 times its pole-to-equator radius), and the London to Los Angeles route curves over Greenland. On a saddle-shaped surface triangles add up to less than 180° (Lobachevsky and Bolyai), as in Escher's Circle Limit prints and Daina Taimina's crochet. Gauss proved curvature can be measured from inside a surface, which is why every flat map lies, and the 1919 eclipse confirmed Einstein's curved space. Measured across the whole universe, the angles still add up to 180°. 0:00 Three right angles 0:26 Euclid's assumption 0:48 Great circles 1:06 The angle excess 1:25 Why you never notice 1:42 Checking the 270° 2:04 Circles on a sphere 2:25 Flight routes 2:40 Hyperbolic geometry 3:23 Gauss's remarkable theorem 3:38 Why every map lies 3:57 Einstein and the flat universe #geometry #sphere #math