The hyperbolic plane is a strange surface. While a sphere, being finite, is smaller than a normal plane (which is infinite), a hyperbolic plane is larger. This is hard to imagine at first. Rucker explains how to think about the hyperbolic surface of pseudosphere:
"To begin to get the notion of a pseudosphere, you might think about crawling around on an endless taffy plane. Every few feet you stop to grab and stretch the plane's material. It gets baggy and wrinkled...more and more spacious." (p. 103)
A pseudosphere is a difficult object to draw, since we usually only have a flat plane available as a drawing surface. (7) Rucker uses a clever method to depict a pseudosphere. He first shrinks an ordinary plane to fit inside a small space, then he does the same with a hyperbolic plane.
It is also possible to illustrate a piece of a hyperbolic plane.
Hyperbolic space is the three-dimensional equivalent of the hyperbolic plane. Hyperbolic space is more roomy than ordinary space, but if you were to visit a hyperbolic universe, you would think that everything in the entire universe was very close to you! How can this be?
Light in hyperbolic space travels in a path that you perceive as curved, though it is actually straight according to the geometry of the space. In order to look at a distant object in the hyperbolic universe, you have to cross your eyes. Your brain is accustomed to images from ordinary space and interprets your crossed eyes as evidence that the object is very close to your face.
Since we do not think everything in our universe is nearby, how can hyperbolic space be a possible shape for our universe? The hyperbolic curvature may be slight enough that we have not yet detected it.
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