Mathematics 300: Geometry
There is no royal road to geometry. --- Euclid to King Ptolemy, who asked if there wasn’t an easier way to learn geometry than working through the axioms and proofs.
Give this man a coin, since he must make a profit from what he learns. --- Euclid to his slave, when a student asked what practical benefit he’d get from learning geometry.
You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone! --- Farkas Bolyai to his son János, who failed to take his father’s advice.
In this course we will carefully cover the first six chapters of the book Euclidean and Non-Euclidean Geometries, by Marvin Jay Greenberg. In addition, we will look at some additional material in Chapter 7; I hope we will have a day or two to discuss the philosophical implications of our study, in Chapter 8.
The course develops a careful axiomatic treatment of plane geometry, following for the most part the work of the famous late nineteenth century German mathematician David Hilbert. We will compare this to the axioms of the ancient Greek mathematician Euclid in his Elements, the most famous mathematical textbook in history.
The last of Hilbert’s (and Euclid’s) axioms is the controversial Parallel Postulate, which asserts (in one of its forms) that through each point not on a line, there exists exactly one line parallel to the original line. We will carefully examine this postulate, from historical, mathematical and philosophical points of view. It turns out that if we add to Hilbert’s postulates the negation of the parallel postulate, we will obtain an extremely different, but equally consistent geometry, called non-Euclidean or hyperbolic geometry. In this geometry there are always infinitely many parallels to a given line through a given point! The creation of this geometry by Gauss, Bolyai and Lobachevsky in the early nineteenth century is one of the most interesting and exciting developments in the history of mathematics.
Since we have a small class, I hope to run this course on the model of a seminar. I will deliver a lecture once or twice a week, to set the groundwork for where we’ve been and where we are going; consequently these lectures will not be on the theorem-proof model, but rather aiming at the "big ideas". The rest of the time, I hope to rely on you students to be active participants. This participation will be of two related types. For most days, I’ll assign to each student either a theorem or topic from the reading to be presented in class, or else an excercise to work out at the board. I will of course be available in the afternoons to talk to you about these individualized assigments. In addition, I’ll pick out some homework exercises as hand-in assignments, due about once a week. This will give you the chance to do some careful mathematical writing.
In order for these approach to work, you as students will need to be active readers of the text! This book is unusually well-written, and so this should not be too difficult. But it requires that (in class and out), you all feel willing to bring out for discussion points in the reading that you fail to understand. For if something is unclear to you, its likely difficult for the rest of us as well! The only difficult part of reading this book is to have a clear understanding of what a given argument stands on. The book follows a carefully axiomatic presentation of neutral, and then hyperbolic geometry, but it also includes a lot of collatoral discussion of arguments that don’t work, arguments appearing in Euclid’s book and elsewhere, etc. Be sure when reading to determine which is which!
The evaluation of your work in the course will be based on your oral work, your hand-in homework, and a written final exam on the last day of the block, one-third for each.
Course goals:
Provisional Course Schedule:
Week One:
Week Two:
Week Three:
Week Four: