Triangles on Curved Surfaces

Flat triangles always sum to 180° — but the world is not flat. On a sphere, a triangle's angles add up to more than 180°; the bigger the triangle, the larger the excess. On a saddle-shaped surface, the angles sum to less than 180°.

This discovery launched non-Euclidean geometry in the 1800s and later underpinned Einstein's general relativity. In that picture, gravity is not a force pulling things together — it is spacetime curving, and objects simply follow the straightest possible paths through that curved geometry.