The importance of the pseudosphere is that it gives a mathematical model for the axioms of non-Euclidean geometry.
Mathematicians worked on non-Euclidean geometry for over 50 years without knowing whether this geometry was consistent. It caused great excitement when the Italian mathematician Beltrami finally proved that non-Euclidean geometry is just as consistent as Euclidean. He used the pseudosphere in his proof.
To this day, there is no proof, acceptable to all mathematicians, that Euclidean geometry itself is consistent.
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