We’re so small in comparison to the Earth that from our point of view, it looks flat. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces—leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Ideas Taking ShapeFor millennia, geometry meant the study of objects in Euclidean space, the flat space we see around us. In Euclidean space, things behave as expected: The shortest distance between any two points is a straight line.