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The Pseudosphere is a very special mathematical object, as is the ordinary sphere. Both have the same gaussian curvature at every point, the ordinary sphere positive, the Pseudosphere negative. This makes this shape look so fascinating at first sight.
The inner mathematical properties are even more interesting. To given an example: One important fact is that the Pseudosphere was the surface used by Beltrami in the 19th century in order to show that a hyperbolic geometry can truely be realized.