Euclidean and Non-Euclidean Geometries

In roughly 300 BCE, Euclid set down his axioms of geometry which subsequently became the standard by which people understood the mathematics of the world around them. In the first half of the 19th century, mathematicians realized, however, that they could remove one of Euclid’s axioms, the one known as the “parallel postulate,” and still produce logically consistent examples of geometries. These new geometries displayed behaviors that were wildly different from Euclidean geometry. In this course we will study examples of these revolutionary non-Euclidean geometries, with a focus on Klein's Erlangen Program, which is a modern way of understanding them. (MATH 0200 or by waiver) 3 hrs. lect.