In this course, we study non-Euclidean geometries (with main focus on hyperbolic geometry) using first the axiomatic approach of Euclid and Hilbert. Hyperbolic geometry is especially counterintuitive (for instance, no matter how long the sides of a triangle are its area cannot exceed a universal constant). Our geometric intuition alone is not enough to predict what the world might look like far away from us.
To discover fascinating facts about non-Euclidean geometries we have to study the global picture. Locally, every geometry can be approximated by Euclidean geometry. This is probably the main reason why we prefer to think of the world around us in terms of Euclidean geometry - this makes calculations easier. The famous Pythagorean theorem holds only in a Euclidean world. Classical models in dimension two are given by Euclidean geometry (geometry of an ideal flat plane), spherical geometry (geometry of the surface of a ball), and hyperbolic geometry, also known as Lobachevskian geometry.
Geometry studies different models of the real world.
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