| Science fiction writers have often pushed past the limits of our senses by exploring the fourth dimension. In the third novel in Cixin Liu's Remembrance of Earth's Past trilogy (the first of which, The Three-Body Problem, was recently made into a Netflix series), a group of humans accidentally crosses into a bubble of four-dimensional space. Suddenly, they find themselves capable of seeing three-dimensional objects the way we see two-dimensional ones — in their entirety, interior and all. When they eventually return to ordinary three-dimensional space, they feel unbearably claustrophobic. Kurt Vonnegut envisioned the fourth dimension very differently in his 1969 novel Slaughterhouse-Five. He writes of an alien race that experiences time as a physical fourth dimension, observing eras and epochs "just the way we can look at a stretch of the Rocky Mountains." To them, past, present and future exist simultaneously; they can travel to different moments in time the way we might fly from one city to another. Since around when science fiction emerged as a genre in the mid-19th century, mathematicians have also been pondering the fourth dimension. You might expect that the more dimensions you add, the weirder shapes get. But mathematically, it turns out that four-dimensional space is the weirdest of them all. Strange new phenomena arise, like "exotic" versions of ordinary space — infinitely many familiar but irrevocably crumpled alternative realities. Techniques break down. Problems, settled for all other dimensions, remain frustratingly open. As one mathematician recently told me: "In dimension 4, everything goes a bit crazy." That's because, according to another, "there's just enough room to have interesting phenomena, but not so much room that they fall apart."
What's New and Noteworthy Research on the fourth dimension really took off with Bernhard Riemann's work in geometry in the mid-1800s. Since then, mathematicians have discovered that four-dimensional shapes are notoriously hard to classify. That's not just because four-dimensional space exhibits unusually exotic behavior, but also because the tools mathematicians use stop working there. One of the biggest breakthroughs in understanding four-dimensional space came in 1981, when a mathematician named Michael Freedman proved the Poincaré conjecture in four dimensions. The conjecture stated that any four-dimensional shape with certain basic properties had to be equivalent to a four-dimensional sphere (the surface of a five-dimensional ball). But Freedman's proof was so difficult and complicated that only a handful of mathematicians could follow it. Even though he had solved one of the biggest problems in the field, the proof risked fading into obscurity. Until a few years ago, that is — when, as Quanta reported, other mathematicians set about unpacking the proof in a 500-page book. An even stronger statement (about whether there are "exotic" spheres in four dimensions, which are equivalent to the ordinary four-dimensional sphere in one way but not in another, stronger way) remains open. In fact, this problem, known as the smooth Poincaré conjecture, has been settled for all dimensions save the fourth. Meanwhile, four-dimensional spheres have continued to surprise. In 2021, for instance, new research demonstrated that they can exhibit many types of unusual symmetries — well beyond the ones we understand intuitively, like rotations and reflections. Spheres aren't alone in getting complicated in four dimensions. I recently wrote an article about new work that complicates the story further. To make sense of four-dimensional shapes, mathematicians often look at the two-dimensional surfaces embedded within them. It turns out that these surfaces can sit inside their four-dimensional homes in far stranger ways than mathematicians had imagined. Even when the math seems like it might get simpler, it doesn't. In an episode of Quanta's podcast The Joy of Why, two mathematicians, Colin Adams and Lisa Piccirillo, explain how in four-dimensional space, all knots can be unraveled, rendered trivial. But instead, we can knot surfaces, like spheres. And the knotted cross sections of those spheres give all kinds of other hints into the weirdness of four-dimensional space. Mathematicians hope to continue exploring the strange realm of the fourth dimension. They've even developed a list of important problems to help guide them on their journey. For now, though, four-dimensional space remains the least understood of all the dimensions — and the most intriguing. | 
