Non-Euclidean Geometry and Nonorientable Surfaces

In the middle part of the nineteenth century, mathematicians first realized that there were different kinds of geometries, geometric systems that did not obey Euclid's rules for plane and solid geometry. The idea that there could be geometries of higher dimensions was disturbing to those philosophers who subscribed to a "realistic" view of geometry, and they were even more upset by the suggestion that there could be different kinds of two-dimensional geometry. Certainly it was well known that the geometry on various surfaces, such as the sphere, was fundamentally different from the geometry of the flat plane. But did the geometry on part of a spherical surface constitute a different two-dimensional geometry, as some mathematicians now claimed? What did it mean to have different kinds of geometry? To explain their ideas, the mathematicians who proposed the new theories resorted to the time-honored device of analogy, and they even asked people to empathize with a two-dimensional creature confined to move along a curved surface. Visualizing dimensions was important in all phases of the controversial new geometries.

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