The Pi-on | Cosmic Variance | Discover Magazine

The Pi-on | Cosmic Variance | Discover Magazine: "But the point of the question is that GR says we don’t live in Euclidean space; we move through a curved spacetime manifold. That’s okay. In a curved space, we could imagine defining the “diameter” of a circle as the maximum geodesic distance connecting two of its points, and taking the ratio of the circumference with that diameter, and indeed it would typically not give us 3.14159… But that doesn’t mean π is changing from place to place; it just means that the ratio of circumference to diameter (defined this way) in a curved space doesn’t equal π. If the circumference/diameter ratio is less than π, you are in a positively curved space, such as a sphere; if it is greater than π, you are in a negatively curved space, such as a saddle. Geometry can also be much more complicated than that, with different ratios depending on how the circle is oriented in space, which is why curvature is properly measured by tensors rather than by a simple number."