The Two Mathematical Perspectives on Shape

By Joseph Howlett

Each week Quanta Magazine explains one of the most important ideas driving modern research. This week, math staff writer Joseph Howlett delves into the interplay of two related but distinct fields — geometry and topology. 

Geometry — the study of shapes — is a staple of every math curriculum. But there is another field of mathematics concerned with shapes, albeit from a very different perspective. Mathematicians use this field, called topology, alongside geometry to probe spaces that defy the imagination.

Geometry involves quantifying properties of fixed shapes: length, angle, volume and more. An object under a geometer’s lens is rigid like a gemstone — it can be moved about but not warped. Topologists, on the other hand, can stretch and compress the shapes they study like clay. They see no difference between a sphere and a cube, because one can easily be molded into the other without breaking or tearing.

Topologists care instead about holes — how many a shape has — and how the shape is wound around itself. A doughnut and a coffee mug both have one hole that prevents them from being shrunk to a point, so they are topologically the same. But they’re different from a sphere or a handle-less coffee cup, which have no holes. Similarly, two knots — formed when shapes are twisted in higher dimensions — are topologically the same if one can be tangled or untangled into the other. They’re different if you can only achieve this by cutting or gluing. All kinds of surfaces, as well as their higher-dimensional relatives called manifolds, have intriguing geometric and topological properties that mathematicians want to make sense of.

To understand the world around us — from the shape of a data set to the shape of our universe — mathematicians are constantly testing the limits of their geometric and topological toolkits.

What’s New and Noteworthy

Often, topologists try to avoid geometry altogether, as when Leonhard Euler proved in 1736 that you can’t traverse the whole city of Königsberg without crossing the same bridge twice. He realized that this problem was really about the path’s topology, not its geometry, in what is now considered the earliest landmark finding in the field.

Since then, mathematicians have used topological methods to solve many problems in geometry. That was the case in a 2020 proof establishing that every smooth, closed curve contains a rectangle. The following year, Quanta reported on the development of sophisticated topological approaches and their application to questions surrounding the geometry of orbiting objects. And last year, Richard Schwartz of Brown University brought in techniques from topology to find geometrically “optimal” shapes, such as the thickest rectangle you can possibly use to make a Möbius strip. In doing so, he verified a conjecture that dates back to the 1970s.

Sometimes the reverse can happen, where understanding the geometry of a shape can give important insights into its topology. It’s long been known, for instance, that local geometric properties such as the so-called curvature of a shape constrain what it can look like topologically. Say that all you know about a given two-dimensional surface is that it’s positively curved at every point. Then it can only be the surface of a sphere or one other, more complicated shape. And geometric curvature data is enough to tell you this.

This year, Quanta wrote about another example of the role curvature plays in determining topology. It turns out that local geometric measurements can even give hints about the topology of the entire universe.

Geometry has plenty of other applications where topology can’t help. This is true any time the objects in question must be rigid, like when you’re packing shapes as tightly as possible, or when distances are critical, like when you’re trying to construct different-size triangles inside a square. But over the last century, it’s really been the relatively nascent field of topology that has helped to broaden its more ancient cousin’s horizons.

Around the Web

• The Royal Swedish Academy of Sciences published a great summary of how the recipients of the 2016 Nobel Prize in Physics used topology in their breakthroughs on phase transitions.
• 3Blue1Brown has a video that visually explains the topology-based proof of the “inscribed rectangle” problem.
• Science News recently ran a story about how the topology of our universe is still up for grabs, and what Pac-Man might be able to tell us about its shape.