Behold the Manifold, the Concept that Changed How Mathematicians View Space

The original version of this story appeared in Quanta Magazine.

Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.

The world is full of such shapes—ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed…