Topology, unlike geometry, ignores precise measurements and focuses on how objects are connected. That makes it the right tool for exploring spaces beyond our three-dimensional experience. In a recent episode of Quanta Magazine's podcast The Joy of Why, mathematician Maggie Miller of the University of Texas at Austin describes what happens when familiar objects meet a fourth spatial dimension.
Miller explains that our intuition about knots fails in 4D. A tangled loop that cannot be loosened in three dimensions can always be pulled apart in four, because the extra dimension gives the string room to slip around itself. But the opposite is true for surfaces: a sphere, which is unremarkable in 3D, can become genuinely knotted in 4D, and that knot cannot be removed without cutting.
To picture such spaces, Miller relies on a method she compares to a reel of film: she imagines a 4D object as a sequence of 3D cross-sections that change over time. Her artistic training, she says, helped her develop these visual strategies. She also discusses how she and her collaborators recently solved a problem about knotted surfaces that Charles Livingston first posed in 1982, showing that even the lowest dimension mathematicians do not fully understand still holds surprises.