Los Angeles Math Circle

3/10 -- High School Circle: Euclid's 5th Postulate, the drama of ideas and the drama of people (Oleg Gleizer)

In this lecture, we will take a look at the murky surroundings of Euclid's 5th Postulate, possibly the most controversial scientific statement of all times. The Great Theorem of Fermat, a proverbial symbol of mathematical complexity, stood open for 358 years. Conjectured by Pierre de Fermat in 1637, it was proven by Andrew_Wiles with the help of his former student Richard Taylor in 1995. The Poincare Conjecture, another very hard math problem made famous outside of the scientific community by its conqueror Gregory Perelman's rejection of the $1,000,000 prize money, was proven in less than 100 years. It took humanity over 2,000 years to realize that the 5th Postulate is indeed a postulate and cannot be derived from other axioms of Euclidean geometry. The breakthrough can only be compared to the Copernican Revolution in astronomy that replaced the geocentric model of the universe with the heliocentric one. Both discoveries broke the millennia-long paradigms. The Copernican Revolution brought about Newtonian mechanics. The discovery of non-Euclidean geometry paved the way to the Relativity Theory of Einstein. In the lecture, we will derive from the axioms the formula for the sum of angles of a triangle in the Euclidean plane and on a sphere. We will see that the first formula, traditionally considered to be the more elementary of the two, is in fact much harder to prove. We will also visit a planet that has only one pole, figure out the shape of the Universe, and some more.