The original version K This story I appeared Quanta Magazine.

Standing in the middle of a field, we can easily forget that we live on a round planet. We are so small compared to Earth that from our perspective, it looks flat.

The world is full of such forms. One that looks flat to ants, although they may have a more complex global structure. Mathematicians call these forms manifolds. Introduced by Bernard Riemann in the mid-19th century, manifolds changed how mathematicians thought about space. It was no longer just a physical setting for other mathematical objects, but an abstract, well-defined object worthy of study in its own right.

This new approach allowed mathematicians to rigorously explore higher-dimensional spaces—giving birth to modern topology, a field devoted to the study of mathematical spaces such as 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 kinds of problems. They are as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchia mathematician at the University of Pisa in Italy. “No, but try learning Russian without learning Cyrillic.”

So, what are manifolds, and what kinds of words do they provide?

Ideas are taking shape

For thousands of years, geometry meant the study of objects in Euclidean space, the flat space we see around us. “Until the 1800s, ‘place’ meant ‘physical space,'” said Jose Freires, a philosopher of science at the University of Seville in Spain.

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

But by the early nineteenth century, some mathematicians had begun to explore other types of geometric spaces—ones that are not flat but curved like a circle or a saddle. In these spaces, parallel lines can eventually meet. The angles of a triangle can add more or less or less than 180 degrees. And doing the calculus can be a lot less straightforward.

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

But some mathematicians wanted to push these ideas further. One of them was Bernhard Riemann, a shy young man who originally intended to study theology – his father was a priest – before being drawn to mathematics. In 1849, he decided to pursue his doctorate under Carl Friedrich Gauss, who was studying the intrinsic properties of curves and surfaces independent of their surroundings.