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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