Physics Asked by kloddant on December 7, 2020
In the 2-D projections of the shape of the universe shown here, we see that the flat universe and the spherical universe are perfectly symmetrical, so any triangle drawn anywhere on them will be the same. However, the hyperbolic universe appears to only be symmetrical on two axes, so any triangle drawn anywhere on it will not necessarily be the same as some identical triangle drawn somewhere else on it. This implies that the hyperbolic universe would have a center and that shapes would have different dimensions at different locations in the universe. Is this the case in this scenario, or is the apparent lack of symmetry an artifact of this being a projection of a 3-D space onto a 2-D image? Or is there something else that I am missing?
It's impossible to draw an accurate picture of a 2D hyperbolic surface, because such a surface cannot be embedded into a 3D euclidean space; this is known as Hilbert's Theorem. The saddle surface in the figure is just an approximation, and serves as an illustration that every point on a hyperbolic surface is a saddle point.
Correct answer by Pulsar on December 7, 2020
You're confusing hyperbolic space with the hyperbolic paraboloid. The hyperbolic paraboloid does not have constant curvature, being the farthest from zero, as you described, in the center of the saddle.
Hyperbolic space is indeed "symmetrical" (homogeneous and isotropic). Standard models for the space are the Poincaré ball, Klein ball, upper half-space, and hyperboloid. Each has its own advantages and disadvantages for visualization and computation.
In the Poincaré ball and upper half space models, straight lines look curved but angles appear correct. In the Klein ball model, straight lines look straight but angles look different from their true measurements. In the hyperboloid model, one requires an additional spacial dimension to embed it. One can find images of these illustrating how hyperbolic symmetry works, but each requires some imagination since our way of perceiving them is Euclidean.
Answered by j0equ1nn on December 7, 2020
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