![]() ![]() The goal is to learn main concepts of hyperbolic geometry in the same manner as we learn high school geometry (in an ideal high school) and to give a rigorous foundation for these concepts (not as in high school, even in an ideal one). 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. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |