But if you take the task seriously, there’s a lot more to it. In fact, one of the largest subdisciplines in mathematics — topology — is devoted exactly to this kind of endeavor, and after centuries of concerted effort, mathematicians aren’t even close to finishing.
Topologists study the properties of general versions of shapes, called manifolds. Their animating goal is to classify them. In that effort, there are a few key distinctions. What exactly are manifolds, and what notion of sameness do we have in mind when we compare them?
Manifolds can be shapes of any dimension, from zero-dimensional points to one-dimensional lines to two-dimensional surfaces (like the surface of a ball) to 100-dimensional spaces (and beyond) that are hard to picture but as mathematically real as anything else. Mathematicians study them because, among other reasons, three- and four-dimensional manifolds provide the setting of our lives.
“They look like where we live, the Earth or space that we live in. Maybe the universe is an interesting manifold,” said Maggie Miller, a postdoctoral fellow at Stanford University.