Dimensions of the Universe

Math Professor Explores the Many Dimensions of the Universe

by Natalie Wilson


While it has been well-known for centuries that Earth is round, little is known regarding the shape of the universe. Jessica Purcell, a faculty member of the Department of Mathematics, is studying such questions using hyperbolic geometry.


“Back in the Middle Ages, people thought the world was flat because if you’re standing on the earth and look around, it looks flat like a two-dimensional plane,” Purcell said. “But even though you’ve got all of these flat pieces, they all connect up to a sphere, which sits in three-dimensional space rather than two-dimensional.”


Planes and spheres are examples of what mathematicians call a manifold. To someone standing within a manifold, it may appear to be a certain dimension. But once one steps out of it, like a person stepping off Earth’s surface into space, the overall shape becomes clearer. 


One may take this concept further by considering whether we truly see the universe as it really is, or whether we are confined to a narrower vision. Those in the Middle Ages saw the earth as a two-dimensional plane, but now we have been able to view it in its three-dimensional state.


"Maybe we envision the universe to continue forever like a three-dimensional plane, when it really connects back on itself, gluing up to form a three-dimensional sphere,” Purcell explained. “If we’re standing in our little space in the universe and we look around, it looks like everything is three-dimensional. You can go up, down, left, right, backward, and forward. But we still don’t know what sort of a three-dimensional manifold the universe is. So what are we really standing in? Does it connect in a higher dimension?”


This question is hard to answer. While we may know how to escape the two-dimensional view of earth and enter outer space, we do not yet know how to exit our universe. Without this convenience, it is extremely difficult to determine which shape the universe should be.

“It’s hard to explore the three-dimensional manifolds because almost all of them, even the simplest, connect up into a higher dimensional universe and we don’t have any way of perceiving that,” Purcell explained. “You can take a ball and be able to identify it as a sphere but you can’t do that for a three-dimensional sphere which sits in four-dimensional space. So that’s one thing that makes these complicated.”


Many discoveries have already been made that shed light on three-dimensional manifolds. Using mathematics, researchers like Dr. Purcell may continue to expand our knowledge of the space we live in.