Euclidean And Non-euclidean Geometries: Develop... -

While Euclidean geometry works for building houses, Non-Euclidean geometry is essential for understanding and the actual shape of our universe.

Think of a saddle or a piece of kale. Through a single point, you can draw infinitely many lines that never touch the original line. Here, the angles of a triangle add up to less than 180° . Euclidean and Non-Euclidean Geometries: Develop...

Think of the surface of the Earth. There are no parallel lines; all "straight" lines (like the equator and longitude lines) eventually cross. In this world, the angles of a triangle add up to more than 180° . Here, the angles of a triangle add up to less than 180°

is the math we all learn in school—the geometry of flat surfaces. It’s based on Euclid’s five postulates, the most famous being the Parallel Postulate : given a line and a point not on it, there is exactly one line through that point that never meets the original line. In this world, the angles of a triangle

challenge that specific rule. They describe curved surfaces where the "shortest path" (a geodesic) behaves differently: