Manifolds

A manifold is a mathematical space that, on a small scale, resembles flat, ordinary space (like a plane), but can have a more complex shape globally. Think of the surface of a sphere: locally, it feels flat when you stand on it, but globally, it's curved and spherical. Manifolds allow us to study complicated shapes and spaces using familiar concepts from geometry and calculus. They are fundamental in fields like physics, engineering, and computer graphics, helping us understand and model diverse structures—from the Earth's surface to the universe's shape—using a consistent mathematical framework.