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What is the shape of the Universe? If you had come along before the 1800s, it likely never would have occurred to you that the Universe itself could even have a shape. Like everyone else, you would have learned geometry starting from the rules of Euclid, where space is nothing more than a three-dimensional grid. Then you would have applied Newton’s laws of physics and presumed that things like forces between any two objects would act along the one and only straight line connecting that. But we’ve come a long way in our understanding since then, and not only can space itself be curved by the presence of matter and energy, but we can witness those effects. It didn’t have to be the case that the Universe, as a whole, would have a spatial curvature to it that’s indistinguishable from flat. But that does seem to be the the Universe we live in, despite the fact that our intuition might prefer it to be shaped like a higher-dimensional sphere. The model of the Universe as: originating from a point, expanding outwards in all directions equally, reaching a maximum size and being drawn back together by gravity, and eventually recollapsing down into a Big Crunch, was one that was preferred by many theoretical physicists throughout the 20th century. But there’s a reason we go out and measure the Universe instead of sticking to our theoretical prejudices: because science is always experimental and observational, and we have no right to tell the Universe how it ought to be. And while “flat” might be the Universe we get, it isn’t some “three-dimensional grid” like you might typically intuit. Here’s what a flat Universe is, as well as what it isn’t. We often visualize space as a 3D grid, even though this is a frame-dependent oversimplification when we consider the concept of spacetime. In reality, spacetime is curved by the presence of matter-and-energy, and distances are not fixed but rather can evolve as the Universe expands or contracts. (Credit: Reunmedia/Storyblocks) In Euclidean geometry, which is the geometry that most of us learn, there are five postulates that allow us to derive everything we know of from them. Travel the Universe with astrophysicist Ethan Siegel. Subscribers will get the newsletter every Saturday. All aboard! Notice: JavaScript is required for this content. Any two points can be connected by a straight line segment. Any line segment can be extended infinitely far in a straight line. Any straight line segment can be used to construct a circle, where one end of the line segment is the center and the other end sweeps radially around. All right angles are equal to one another, and contain 90° (or π/2 radians). And that any two lines that are parallel to each other will always remain equidistant and never intersect. Everything you’ve ever drawn on a piece of graph paper obeys these rules, and the thought was that our Universe just obeys a three-dimensional version of the Euclidean geometry we’re all familiar with. But this isn’t necessarily so, and it’s the fifth postulate’s fault. To understand why, just look at the lines of longitude on a globe. This diagram of a globe is centered on the prime meridian, which is our arbitrary definition of 0 degrees longitude. Lines of latitude are also shown. On a flat surface, parallel lines never intersect, but this is not true on a sphere. At the equator, all lines of longitude are parallel, but all those longitudinal lines also cross in two places: at the north and south poles. (Credit: Hellerick/Wikimedia Commons) Every line of longitude you can draw makes a complete circle around the Earth, crossing the equator and making a 90° angle wherever it does. Since the equator is a straight line, and all the lines of longitude are straight lines, this tells us that — at least at the equator — the lines of longitude are parallel. If Euclid’s fifth postulate were true, then any two lines of longitude could never intersect. But lines of longitude do intersect. In fact, every line of longitude intersects at two points: the north and south poles. The reason is the same reason that you can’t “peel” a sphere and lay it out flat to make a square: the surface of a sphere is fundamentally curved and not flat. In fact, there are three types of fundamentally different spatial surfaces. There are surfaces of positive curvature, like a sphere; there are surfaces of negative curvature, like a horse’s saddle; there are surfaces of zero curvature, like a flat sheet of paper. If you want to know what the curvature of your surface is, all you have to do is draw a triangle on it — the curvature will be easier to measure the larger your triangle is — and then measure the three angles of that triangle and add them together. The angles of a triangle add up to different amounts depending on the spatial curvature present. A positively curved (top), negatively curved (middle), or flat (bottom) Universe will have the internal angles of a triangle sum up to more, less, or exactly equal to 180 degrees, respectively. (Credit: NASA/WMAP Science Team) Most of us are familiar with what happens if we draw a triangle on a flat, uncurved sheet of paper: the three interior angles of that triangle will always add up to 180°. But if you instead have a surface of positive curvature, like a sphere, your angles will add up to a greater number than 180°, with larger triangles (compared to the sphere’s radius) exceeding that 180° number by greater amounts. And similarly, if you had a surface of negative curvature, like a saddle or a hyperboloid, the interior angles will always add up to less than 180°, with larger triangles falling farther and farther short of the mark. This realization — that you can have a fundamentally curved surface that doesn’t obey Euclid’s fifth postulate, where parallel lines can either intersect or diverge — led to the now-almost 200 year old field of non-Euclidean geometry. Mathematically, self-consistent non-Euclidean geometries were demonstrated to exist independently, in 1823, by Nicolai Lobachevsky and Janos Bolyai. They were further developed by Bernhard Riemman, who extended these geometries to an arbitrary number of dimensions and wrote down what we know of as a “metric tensor” today, where the various parameters described how any particular geometry was curved.