How is 3-sphere curvature measured? If a 2-D being living "in" the surface of a sphere tried to measure the 3-D curvature of the sphere, how would they go about it? They couldn't detect the curvature by looking for curvature in the paths of signals, because if the surface of their sphere was as transparent, isotropic, and homogeneous as our universe is, then wouldn't any signal they emit show no deviation from a geodesic when viewed on a cosmological scale, and we would see any 4-D geodesic in our universe as a straight line, the path followed by an "unperturbed" particle? The only way I can think of to detect higher-dimensional curvature would be to examine the effect on particles which have traveled over a cosmological distance. Since particles have non-zero dimension, then absent some kind of resistance by the particle or a field, they would tend to be stretched out more and more the further they traveled. (Two sides of the particle on opposite sides of it's geodesic path would tend to follow slightly different geodesics in the surface of a 3-sphere, leading to a "spreading out" of the particle as it traveled, assuming, again, that the particle or a field didn't counteract this somehow.) I remember from a quantum mechanics course I took in college that the wavefunctions of photons tend to spread out more and more the further they travel. I assumed, however, that this phenomenon was explained by quantum mechanics and didn't require curvature of space in higher dimensions. Please don't respond with relativistic tests, as relativity doesn't deal with 4 spatial dimensions; it deals with 3 spatial dimensions and 1 time dimension.