Hyperbolic Geometry: Four similarities, one big difference.

300 BC. Euclid wrote his masterpiece ‘Elements’. In here he spoke about 5 postulates, which should form the base of geometry. Since the fifth differs from the first four, people though that this parallel postulate should be a theorem instead of a postulate. If this is true, they should be able to prove the fifth one by the first four. Over two thousand years people tried to find such a prove, but they didn’t manage to find one. Till the three mathematicians Gauss, Lobachevsky and Bolyai independently invented hyperbolic geometry. With this new geometry, they proved the independence of the fifth postulate. In my presentation I will explain the main topics of this geometry utilizing the model M.C. Escher used for his ‘Angles and Demons’. In addition I will discuss three other models and I will show that these models are equivalent. Finally you will see how Escher used this geometry in some of his beautiful works of art.