Original version With 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 are so miniature compared to the Earth that from our point of view it appears flat.
The world is full of such shapes – ones that appear flat to the ant living on them, even though they may have a more complicated global structure. Mathematicians call these shapes manifolds. The manifolds introduced by Bernhard Riemann in the mid-19th century changed the way 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 novel perspective enabled mathematicians to rigorously study high-dimensional spaces, leading to the birth of up-to-date topology, a field devoted to the study of mathematical spaces such as manifolds. Manifolds have also come to play a key 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?” he said Bianchi Factorymathematician at the University of Pisa in Italy. “No. But try learning Russian without learning Cyrillic.”
So what are manifolds and what kind of vocabulary do they provide?
Ideas take shape
For millennia, geometry has meant the study of objects in Euclidean space, the flat space we see around us. “Until the 19th century, ‘space’ meant ‘physical space,'” said José Ferreirós, a philosopher of science at the University of Seville in Spain – the equivalent of a line in one dimension or a flat plane in two dimensions.
In Euclidean space, everything behaves as expected: the shortest distance between any two points is a straight line. The angles of a triangle add up to 180 degrees. The tools of calculus are strong and well-defined.
However, in the early 19th century, some mathematicians began to explore other types of geometric spaces – those that are not flat, but rather curved, like a sphere or a saddle. In these spaces, parallel lines can eventually intersect. The angles of a triangle can add up to more or less than 180 degrees. And doing calculations can become much less effortless.
The mathematical community had difficulty accepting (or even understanding) this shift in geometric thinking.
However, some mathematicians wanted to push these ideas even further. One of them was Bernhard Riemann, a shy youthful man who originally planned to study theology – his father was a pastor – before becoming interested in mathematics. In 1849, he decided to pursue his doctorate under Carl Friedrich Gauss, who studied the inherent properties of curves and surfaces, independent of the space surrounding them.