Reality Is Not What It Seems: The Journey to Quantum Gravity

by Carlo Rovelli

Now, think about it; for someone who has learned from his teacher that the form of the surface of our planet is such that by walking always in a straight line we return to the point we started from, it is perhaps not so difficult to take the next obvious step and imagine that the form of the entire universe is such that, flying always in a straight line, we return to the same point of departure: a 3-sphere is a space in which “two winged knights who could fly in opposite directions would meet up on the other side.” In technical terms: the description of the geometry of Earth offered by Brunetto Latini in Li tresor is given in terms of intrinsic geometry (seen from the inside) rather than extrinsic (seen from the outside), and this is exactly the one that is suitable to generalize the notion of “sphere” from two dimensions to three. The best way of describing a 3-sphere is not to try to “see it from the outside” but rather to describe what happens when moving within it. The method developed by Gauss to describe curved surfaces, and generalized by Riemann to describe the curvature of spaces in three or more dimensions, basically amounts to Brunetto Latini’s way. That is to say, the idea is to describe a curved space not as “seen from the outside,” stating how it curves in an external space, but instead in terms of what may be experienced by somebody within that space, who is moving and always remaining within it. For instance, the surface of an ordinary sphere, as Brunetto o...