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FYSE 1423 - The Story of 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 ’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 postulate. For years, mathematicians believed that the 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 , 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 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 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.

If you choose to write about a specific person, your paper should balance biographical information about the mathematician with an explanation of a part of the mathemati- cian’s work that relates to geometry. Your biographical data should include information about the time period in which the mathematician lived in addition to facts about your specific mathematician. When discussing the mathematician’s work, be sure to put his research into context by explaining what was known before him and how his contribu- tions changed the landscape of geometry. You do not need to understand the details of the mathematician’s work, but aim to provide an accessible summary that would make sense to an intelligent lay person who knows about, for example, Euclid’s Elements.

If you choose to write about a period of development, describe the topic through a historical lens. What was the state of a↵airs in geometry before the time period in question, what were the significant new ideas that came about during the development, and how were things di↵erent after that period? Bring the topic to life by writing a bit about the mathematicians involved. You will necessarily include less about the lives of the specific mathematicians than you would if you were writing a biography, but you should still aim to give a sense of the personalities involved and how the mathematicians interacted with each other.

When I read your work, I will be looking for a central theme that unifies your work. It does not have to be a complicated theme (often simple themes are best), but it should create the thread that ties together what you write in your paper. Please also focus on clarity of writing and organization, both on the level of individual paragraphs and on the level of your paper as a whole. Here is a list of geometers and historical periods of development to choose from. You are not required to choose your topic from one of these lists but if you would like to choose a topic that is not on this list, please get prior approval from me.

Archimedes Khayyam Descartes Legendre J.Bolyai Lobachevsky Gauss Riemann Beltrami Klein Poincar´e Hilbert Mathematics in ancient Failed attempts at proving the parallel postulate The Poincar´econjecture

Dates Your mostly polished draft is due on Monday, October 27. Please submit an • electronic pdf copy of your draft to the Dropbox folder for our class on the Classes server by 5pm on that day. IwillreadyourdraftsandgetthembacktoyouwithcommentsbyMonday, • November 3. Your revised paper is due back to me via the Dropbox folder by 5pm on Monday, November 10. 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 October 20-26. 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.

Notes about sources Please use at least three (preferably more) sources in preparing your paper. We • will use the Chicago Manual of Style format for citations. Because Wikipedia is an unreliable source, you may not use it as one of your sources • for your written paper. In general, be wary of internet sources. You may use them, but before you do, consider carefully the source and whether it is reliable. Please write and (electronically) the honor pledge at the bottom of your paper • before turning it in. FYSE 1423: The Story of Geometry Fall 2014

Third Paper Assignment: Euclid’s Legacy

In a 4-6 page (double spaced, 12pt font) paper, please address the following question: in what ways did Euclid’s work in Elements set the stage for the development of modern geometry? Phrased another way: how did modern geometry develop from Euclid’s work? Iamnotlookingforamerehistoricalaccountingofwhichresultscamefirst,whichcame second, and so on. Rather, I am looking for your interpretation of how all of the math that we have studied this semester fits together, theoretically as well as historically. As we all know, Euclid’s Elements is a very important work, but it is also not without flaws. In your paper, please include discussion of both the strengths and the weaknesses of Elements.

Astrongpaperwillcommentonalloftheperiodsofhistoricaldevelopmentthatwe have seen this semester, and may possibly push beyond what we have seen to discuss other related topics. You will likely draw from the readings that we have done for class, and you may draw from the work in your did in your independent research papers as well. While you are not required to find other outside sources for your paper, carefully chosen supporting material can strengthen and add depth to your work.

When I read your work, I will be looking for a central thesis that unifies your work, and Iwillbepayingattentiontohowyousupportyourthesisthroughoutyourwork.Your aim should be an orderly and engaging development of your discussion. To the extent that you use outside sources, please make sure to paraphrase and cite correctly, as well as comment on how the source supports your argument. Dates

Your mostly polished draft is due on Monday, November 24. Please submit an • electronic pdf copy of your draft to the Dropbox folder for our class on the Classes server by 5pm on that day.

IwillreadyourdraftsandgetthembacktoyouwithcommentsbyMonday, • December 1. Your revised paper is due back to me via the Dropbox folder by 5pm on Wednesday, December 10.

Each of you will meet with our peer mentor, Kate Eiseman, in the process of • writing your paper. You may meet with her at whatever stage in your writing suits you best. 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.

Notes about sources

Please use the author-date format of the Chicago Manual of Style for citations. • You may not use Wikipedia as a source for you paper. Library sources are best. • You may use information you find on the internet but before you do, consider carefully the source and whether it is reliable.

Please write and sign (electronically) the honor pledge at the bottom of your paper • before turning it in.