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 how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.

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. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis, and physics.

Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”

So what are manifolds, and what kind of vocabulary do they provide?

Ideas Taking Shape

For millennia, geometry meant the study of objects in Euclidean space, the flat space we see around us. “Until the 1800s, ‘space’ meant ‘physical space,’” said José Ferreirós, a philosopher of science at the University of Seville in Spain—the analogue of a line in one dimension, or a flat plane in two dimensions.

In Euclidean space, things behave as expected: The shortest distance between any two points is a straight line. A triangle’s angles add up to 180 degrees. The tools of calculus are reliable and well defined.

But by the early 19th century, some mathematicians had started exploring other kinds of geometric spaces—ones that aren’t flat but rather curved like a sphere or saddle. In these spaces, parallel lines might eventually intersect. A triangle’s angles might add up to more or less than 180 degrees. And doing calculus can become a lot less straightforward.

The mathematical community struggled to accept (or even understand) this shift in geometric thinking.

But some mathematicians wanted to push these ideas even further. One of them was Bernhard Riemann, a shy young man who had originally planned to study theology—his father was a pastor—before being drawn to mathematics. In 1849, he decided to pursue his doctorate under the tutelage of Carl Friedrich Gauss, who had been studying the intrinsic properties of curves and surfaces, independent of the space surrounding them.