FYSE 1423 - The Story of Geometry Fall 2014 Emily Proctor [email protected] 310 Warner Hall Office Hours: Tu 11-12, W 11-12, Th 2:30-4:00, and by appointment Peer Mentor: Kate Eiseman [email protected] Librarian: Bryan Carson [email protected] Course Description In this course we will look at one storyline in the development of modern geometry. We will begin the semester with an examination of Euclid's Elements. Written over 2000 years ago, Elements is one of the most influential textbooks ever, and has long been held up as a shining example of deductive reasoning. While this book deserves great praise, we will see that it is not without flaws. After familiarizing ourselves with Elements, we will investigate its problems, and see how modern mathematicians, Hilbert in particular, shored up Euclid's work years later. Euclid's work is based on five postulates, the last of which is often referred to as the parallel postulate. For years, mathematicians believed that the parallel postulate could be deduced from the first four postulates, and gave many erroneous proofs that it was. Finally, in the 19th century, three mathematicians, Gauss, Bolyai, and Lobachevsky, came to the outstanding revelation that there are consistent geometric systems for which the first four postulates hold but for which the parallel postulate does not. This discovery shook the foundations of geometry and opened the door to modern geometry. We will examine one such geometric system, and at the end of the semester, we will learn about how modern geometers view the subject. Modern geometry does not look much like Euclid's Elements, but we will see how its roots come from Euclid's ancient work. Course Website The website for this course can be found at: http://f14.middlebury.edu/fyse1423a Look here for information about the course as well as homework and writing assignments. Texts • Euclid's Elements, edited by Dana Densmore, translated by T. L. Heath. • Geometry: Euclid and Beyond, by Robin Hartshorne. • The Shape of Space, second edition, by Jeffrey Weeks. 1 Homework We will have homework for each class meeting. Some of the assignments will be \traditional" math homework, in the form of problems to work out, and some of our homework will be reading and writing assignments. Please check the website for our daily assignments. Although you will not always have something to turn in, it is essential that you do the assignment each night as our activities in class will depend heavily on the work that you do between classes. Major Assignments Along with smaller assignments, there will be three main papers for this course. For each paper, you will need to produce both a polished draft, along with a substantial revision of that draft. For your planning purposes, the tentative due dates for the polished drafts are Monday, September 29 Monday, November 3 Monday, December 1 and the revisions would be due roughly two weeks later. I will pass out more specific information about the assignments and due dates during the semester. Please note that I have a very strict policy of not accepting late work without penalty, so make sure to plan accordingly. Support We have both a peer mentor, Kate Eiseman, and a librarian, Bryan Carson, assigned to our class to help support you in your transition to Middlebury. Our peer mentor is available to help with your writing process, as well as with your oral presentation and time-management skills. At some points in the semester, I will require that you meet with Kate to discuss your writing, but she is available and ready to help even if I do not require that you speak with her. She may also hold a workshop on oral presentation skills for you, depending on the interest of the class. Our librarian is available to help you learn to navigate the Middlebury College library. He will help acquaint you with the resources available through the library and can also help with specific questions you have about finding library materials. Both Kate and Bryan's contact information is listed at the top of the syllabus, so please do not hesitate to contact either of them with questions. Honor Code The Middlebury College Honor Code is central to the academic community that we foster at Middlebury. To make sure that you get started on the right foot with your college career, in class we will discuss the Honor Code as it applies to your life at Middlebury College in general. For this particular course, the Honor Code applies as follows: your written work and any homework problems you turn in must be entirely your own 2 work, and may not be work that you have turned in for another class prior to ours. Please properly cite any sources that you reference in your papers. For any homework problems, please cite any help that you have received from a classmate. Online sources are not allowed for solving homework problems. If you have any questions at any point, please ask me! Attendance I expect to attend every class during the term and to participate fully during class. Please on time, and be ready to start working at the time that class starts. I will not take formal attendance but I will notice if you are gone. Depending on the circumstances, missing class could have a negative impact on your final grade. If you are seriously ill for a number of days, contact me and your dean as soon as possible so that we can work with you to help you keep up with the rest of the class. Grading I will determine final grades according to the following percentages: First Major Paper 30% Second Major Paper 30% Third Major Paper 30% Active Class Participation and Smaller Assignments 10% 3 FYSE 1423: The Story of Geometry Fall 2014 First Paper Assignment: Book IV of Euclid’s Elements. Your task for this assignment is to explore Book IV of Euclid’s Elements.Iwouldlike you to read and work to understand all of the definitions, all of the statements of the sixteen propositions in Book IV, and as many proofs of the propositions as you would like. Once you have done this, please write a 4-6 page paper that summarizes what you have found. In your paper, you should include a summary and analysis of the definitions and the propositions. You aim here should be to try to explain to an intelligent lay person your observations about the contents of Book IV. In other words, you should read Euclid’s work, internalize it, and explain it in your own language. In order to give some depth to your work, I would like you to read at least one proof carefully and incorporate the proof, again in your own words, into your paper. For this, please choose one of the proofs of Proposition 10, Proposition 11, or Proposition 12. When you write about the proof, first summarize the main steps of the proof and then work through the reasoning in finer detail. Whenever Euclid makes reference to an earlier proposition in his proof, please include in your exposition a statement, in your own words, of what the earlier proposition says. If Euclid leaves out any references, please supply them. Again, you should think of your audience here as an intelligent lay person: someone who has basic background in mathematics but who is not necessarily an expert in geometry. Note that although you do not need to read all of the proofs in Book IV, the more proofs you read, the better your paper is likely to be. Even if you don’t incorporate all of the proofs into your paper, knowing them will give you a fuller and more nuanced understanding of the work. Dates: Your mostly polished draft is due in class on Monday, September 29. Please also • email me an electronic copy by 5pm on that day. IwillreadyourdraftsandgetthembacktoyouwithcommentsbyMonday, • October 6. Your revised paper is due back to me in class on Wednesday, October 15. Please also email me an electronic copy by 5pm on that day. Each of you will meet with our peer mentor, Kate Eiseman, in the process of • writing your initial draft of your paper. She will email the class a list of available meeting times during September 17-28. Please set up a meeting with her and send her a copy of your current draft 24 hours before your meeting so that she has time to read it before talking with you. A note about sources: As you know, Euclid’s Elements is an ancient text and many, many people have written about it over the years. Although there may be other sources out there that describe the work in Book IV of Elements,forthispaperIaminterestedinyour interpretation and explanation of Book IV. Thus, the only resources you may use for this paper are your copy of Elements; your copy of Hartshorne (Chapter 1, Section 4 in particular might be worth a read); and conversations with each other, Kate Eiseman, and me. FYSE 1423: The Story of Geometry Fall 2014 Second Paper Assignment: Geometry in Context Your task for this assignment is to research either the life and work of a mathematician who was influential in the development of geometry or a period of major development in the history of geometry. Write about your findings in a 4-6 page (double spaced, 12pt font) paper. The goal of this assignment is to give some mathematical and historical context as well as life and a human side to the material we are studying in class.