Listed in: Mathematics and Statistics, as MATH-255
David A. Cox (Section 01)
In Euclidean geometry, the parallel axiom asserts that if we have a line and a point not on the line, then there is a unique line through the point which is parallel to the given line. This seemingly obvious statement has many consequences, including the Pythagorean Theorem and the fact that the angles of a triangle sum to 180 degrees. In the nineteenth century, it was discovered that this is not the only possible geometry. The course will begin with neutral geometry, which makes no assumptions about parallel lines. We will then study non-Euclidean geometry, which uses a different parallel axiom. Familiar objects like circles and triangles behave differently in this geometry. For example, rectangles don't exist, and the angles of a triangle sum to less than 180 degrees, and the difference is proportional to the area of the triangle. This will allow us to construct an eight-sided house where every corner is a right angle. Besides proving some fun theorems, we will also study the history of non-Euclidean geometry. The final part of the course will be an introduction to differential geometry. The key concepts will be geodesics (which generalize straight lines) and curvature (which measures how the space is warped). This will allow us to make models of non-Euclidean geometry and explore how geometric ideas apply in a much wider context. Four class hours per week.
Requisite: Math 211 or consent of the instructor. Fall semester. Professor Cox.