To get around these difficulties, astronomers generally look not for copies of ourselves but for repeating features in the farthest thing we can see: the cosmic microwave background (CMB) radiation left over from shortly after the Big Bang. And maybe they’re all too far away for us to see anyway. Making matters worse, different copies of yourself will usually be different distances away from you, so most of them won’t look the same as each other. Maybe we’re seeing unrecognizable copies of ourselves out there. For one thing, they all have the same local geometry as Euclidean space, so no local measurement can distinguish among them.Īnd if you did see a copy of yourself, that faraway image would show how you (or your galaxy, for example) looked in the distant past, since the light had to travel a long time to reach you. Even so, it’s surprisingly hard to rule out these flat shapes. When we look out into space, we don’t see infinitely many copies of ourselves. Is Our Universe One of These Other Flat Shapes? In each of these worlds there’s a different hall-of-mirrors array to experience. There are also flat infinite worlds such as the three-dimensional analogue of an infinite cylinder. The three-dimensional torus is just one of 10 different flat finite worlds. But the changes we’ve made to the global topology by cutting and taping mean that the experience of living in the torus will feel very different from what we’re used to.įor starters, there are straight paths on the torus that loop around and return to where they started: Since the geometry of this universe comes from a flat piece of paper, all the geometric facts we’re used to are the same as usual, at least on a small scale: Angles in a triangle sum to 180 degrees, and so on. Imagine you’re a two-dimensional creature whose universe is a flat torus. But we can reason abstractly about what it would feel like to live inside a flat torus. Inside ordinary three-dimensional space, there’s no way to build an actual, smooth physical torus from flat material without distorting the flat geometry. But this stretching distorts lengths and angles, changing the geometry. You’d have to use some stretchy material instead of paper. Making the cylinder would be easy, but taping the ends of the cylinder wouldn’t work: The paper would crumple along the inner circle of the torus, and it wouldn’t stretch far enough along the outer circle. If you actually tried to make a torus out of a sheet of paper in this way, you’d run into difficulties. We cheated a bit in describing how the flat torus works. Now, you might be thinking, “This doesn’t look flat to me.” And you’d be right. The simplest example of a flat three-dimensional shape is ordinary infinite space - what mathematicians call Euclidean space - but there are other flat shapes to consider too. The angles of a triangle add up to 180 degrees, and the area of a circle is π r 2. This is the geometry we learned in school. Let’s explore these geometries, some topological considerations, and what the cosmological evidence says about which shapes best describe our universe. Only three geometries fit this description: flat, spherical and hyperbolic. The local fabric of space looks much the same at every point and in every direction. The other is about its topology: how these local pieces are stitched together into an overarching shape.Ĭosmological evidence suggests that the part of the universe we can see is smooth and homogeneous, at least approximately. One is about its geometry: the fine-grained local measurements of things like angles and areas. We can ask two separate but interrelated questions about the shape of the universe. Just as the sphere offered an alternative to a flat Earth, other three-dimensional shapes offer alternatives to “ordinary” infinite space. But most of us give little thought to the shape of the universe. Today, we know the Earth is shaped like a sphere. There was a time, after all, when everyone thought the Earth was flat, because our planet’s curvature was too subtle to detect and a spherical Earth was unfathomable. That’s our mental model for the universe, but it’s not necessarily correct. When you gaze out at the night sky, space seems to extend forever in all directions.
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