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

The archetypal version of this story appeared in Quanta Magazine.

Standing successful the mediate of a field, we tin easy hide that we unrecorded connected a circular planet. We’re truthful tiny successful examination to the Earth that from our constituent of view, it looks flat.

The satellite is afloat of specified shapes—ones that look level to an ant surviving connected them, adjacent though they mightiness person a much analyzable planetary structure. Mathematicians telephone these shapes manifolds. Introduced by Bernhard Riemann successful the mid-19th century, manifolds transformed however mathematicians deliberation astir space. It was nary longer conscionable a carnal mounting for different mathematical objects, but alternatively an abstract, well-defined entity worthy studying successful its ain right.

This caller position allowed mathematicians to rigorously research higher-dimensional spaces—leading to the commencement of modern topology, a tract dedicated to the survey of mathematical spaces similar manifolds. Manifolds person besides travel to inhabit a cardinal relation successful fields specified arsenic geometry, dynamical systems, information analysis, and physics.

Today, they springiness mathematicians a communal vocabulary for solving each sorts of problems. They’re arsenic cardinal to mathematics arsenic the alphabet is to language. “If I cognize Cyrillic, bash I cognize Russian?” said Fabrizio Bianchi, a mathematician astatine the University of Pisa successful Italy. “No. But effort to larn Russian without learning Cyrillic.”

So what are manifolds, and what benignant of vocabulary bash they provide?

Ideas Taking Shape

For millennia, geometry meant the survey of objects successful Euclidean space, the level abstraction we spot astir us. “Until the 1800s, ‘space’ meant ‘physical space,’” said José Ferreirós, a philosopher of subject astatine the University of Seville successful Spain—the analogue of a enactment successful 1 dimension, oregon a level flat successful 2 dimensions.

In Euclidean space, things behave arsenic expected: The shortest region betwixt immoderate 2 points is simply a consecutive line. A triangle’s angles adhd up to 180 degrees. The tools of calculus are reliable and good defined.

But by the aboriginal 19th century, immoderate mathematicians had started exploring different kinds of geometric spaces—ones that aren’t level but alternatively curved similar a sphere oregon saddle. In these spaces, parallel lines mightiness yet intersect. A triangle’s angles mightiness adhd up to much oregon little than 180 degrees. And doing calculus tin go a batch little straightforward.

The mathematical assemblage struggled to judge (or adjacent understand) this displacement successful geometric thinking.

But immoderate mathematicians wanted to propulsion these ideas adjacent further. One of them was Bernhard Riemann, a shy young antheral who had primitively planned to survey theology—his begetter was a pastor—before being drawn to mathematics. In 1849, helium decided to prosecute his doctorate nether the tutelage of Carl Friedrich Gauss, who had been studying the intrinsic properties of curves and surfaces, autarkic of the abstraction surrounding them.