ABSTRACT
What is the shape of the universe? For a long time, people thought that it had to be the infinite three-dimensional space described by Euclid and therefore called it Euclidean 3-space. Then a few people realized that that wasn’t the only possibility. It might, for instance, be like the surface of a sphere in four-dimensional space, which nowadays is called three-dimensional spherical space or just the 3-sphere. Then, it would be “finite but unbounded,” meaning that the total volume would be finite even though there would be no bounding walls. In that case, space would be curved, in a three-dimensional analog to the two-dimensional surface of the Earth, which has finite total area even though there is no edge for people to fall off. Still more recently, physicists have become aware that the universe might not only be finite and unbounded, but also exactly flat, in the sense that the local geometry is exactly Euclidean.
