Look at the manifold, the concept that changed the way mathematicians look at space

The original version to This story Featured in Quanta Magazine.

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

The world is full of such shapes, ones that would appear flat to an ant living on them, even though they may have a more complex world structure. Mathematicians call these shapes manifolds. Manifolds, introduced by Bernhard Riemann in the mid-19th century, changed the way mathematicians thought about space. It is no longer just a physical setting of other mathematical objects, but an abstract, well-defined object worthy of study in its own right.

This new perspective allowed mathematicians to explore higher-dimensional spaces with precision, 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 engineering, dynamical systems, data analysis, and physics.

Today, they provide mathematicians with a common vocabulary for solving all kinds of problems. It is as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” He said Fabrizio Bianchia 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 take shape

For thousands of years, geometry has meant studying things in Euclidean space, the flat space we see around us. “Until the 19th century, the term ‘space’ meant ‘physical space,’” which is analogous to a line in one dimension, or a flat plane in two dimensions, said José Ferreros, a philosopher of science at the University of Seville in Spain.

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

But by the early 19th century, some mathematicians began to explore other types of geometric spaces, spaces that are not flat but curved like a sphere or a saddle. In these spaces, parallel lines may eventually intersect. The sum of the angles of a triangle may be more or less than 180 degrees. And doing calculus can become much less straightforward.

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

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