One of the reasons for the demise of Euclidean geometry is that it has been subsumed into algebra. Once you introduce coordinates, you can translate the theorems of geometry into theorems about numbers, or sets of numbers, and can prove them like that. This procedure can often result in ugly proofs, the price one pays for not viewing a geometrical problem geometrically.

This course begins with spherical geometry, that is, the two-dimensional geometry that results from working on the surface of a sphere rather than in a plane. (There is a very obvious reason for wanting to study this.) It is not impossible to devise coordinate systems for the sphere. For example, one can use longitude and latitude, or one can think of the sphere as the set of all points (x,y,z) in R^3 such that x^2+y^2+z^2=1. However these systems have disadvantages: longitude is not well-defined or continuous at the poles, and using three coordinates for a two-dimensional set is not ideal. Thus, the ugliness (and difficulty) of using coordinates is greater than in the plane. This forces one to think geometrically.

Hyperbolic geometry, the most important topic of the course, is even more troublesome, because not only does the hyperbolic plane not have a natural coordinate system, one cannot even regard it as a subset of R^3 without distorting it. Once again, one is forced to think geometrically, but that is only the beginning of the difficulty. For other difficulties peculiar to hyperbolic geometry click here.

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