Math 32: The Shape of Space
Winter 2012

Instructor Office Hours Course Description Learning Outcomes Course Reading/Textbook Grades Daily Schedule and HW Final Projects Student Needs Honor Code

Homework Page


Patricia Cahn
Office: Kemeny 243
Email: first dot last at dartmouth dot edu

Office Hours

Tues 3-4, 5:10-5:50
Wed 3-4, 5-6
Thurs 10-12
By appointment.

I also respond to emails the day I receive them as long as I receive them before 10 pm.

Course Description

In this course, we will examine various concepts in topology and geometry, including knots, polyhedra, surfaces, orbifolds, and wallpaper patterns.  There are several themes that run throughout this course.  One theme is the idea of an invariant.  Invariants are mathematical tools for telling objects apart.  For example, take two pipe cleaners and bend them into the knots shown above.  Can you transform one to the other if the pipe cleaner is not allowed to pass through itself?  If so, we say these knots are the same, or more precisely, isotopic.  If not, we can try to use invariants to prove that they are different.  

In this course, we will use many of the activities developed for the course Geometry and the Imagination,  designed by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston (see Course Reading/Textbook).  If you are thinking about taking this class and would like to get a sense of what we'll be doing, this document is a good place to look.

Learning Outcomes

Learning outcomes are things that you will be able to do at the end of the term as a result of taking this course.  I write outcomes so that I can make sure that all of the work I'm asking you to do has a clear relationship to my goals for you, but I thought you might be interested in reading them as well.  

1. Students will develop their ability to visualize topological and geometrical objects through in-class activities involving physical models of these objects.

2. Students will learn to distinguish topological and geometric objects, including knots and surfaces, using invariants.
3. Students will recognize and name the 17 wallpaper patterns.
4. Students will gain confidence in their ability to solve problems without being provided with a method of approach in advance.
5. Students will gain confindence in their ability to read a mathematical text.

6. Students will write clearly about mathematical ideas.

Course Reading/Textbook

Much of the material for this course will come from the document Geometry and the Imagination, available here.
Our official textbook is The Shape of Space, Second Edition, by Jeffrey Weeks.  Your reading assignments will come from this book.
I will also use some material from Prasolov and Sossinksy's Knots, Links, Braids, and 3-Manifolds, but you do not need to buy this book.

Daily Schedule and Homework

The link to our daily schedule is at the top of the page.  Here is roughly what we will cover each week:

Week 1: Regular homotopy theory.  Turning a sphere inside out. 
Weeks 2-3: Knot theory: knot diagrams, Reidemeister moves, 3-colorings, the Kauffman bracket, and the Jones polynomial.
Week 4: Polyhedra (including the platonic solids), maps, Euler's formula, planar graphs, graphs on surfaces, cell decompositions of surfaces.
Week 5: Intrinsic and extrinsic topology and geometry of surfaces.  Connected sums.  Orientability.  Euler characteristic.  Classification of surfaces using Conway's ZIP proof.
Week 6-7: Wallpaper and orbifolds: types of symmetries, pattern identification strategies.
Week 7-8: Geometry of surfaces: elliptic and hyperbolic geometry, the Gauss-Bonnet formula, hyperbolic and spherical wallpaper.
Week 9: 3-Manifolds: What are they, and some examples


25% Homework (Outcomes 2, 3, 4)

There will be a homework assignment due about once a week.  There will also be some reading assignments.

Late homework will be accepted up to 2 days after the due date for half credit.  After that, late homework will not be accepted.  

25% Quizzes (Outcomes 2, 3, 5)

There will be about four quizzes throughout the course.  If you need to miss a quiz (for a good reason), you will be allowed to take it at another time, but you must let me know in advance of the quiz date.

5% Class Participation (Discussion board, class attendance, etc) (Outcome 1)

Since classroom activities are a major component of the course, attendance is required.
If you cannot make it to class, please send me an email in advance.

15% Course Portfolio (Outcomes 2, 3,  6)

You will demonstrate your big-picture understanding of the course material through a course portfolio.  I will collect it twice, once around the middle of term and once near the final.  

I will provide more details later, but the portfolio will definitely contain the following components:

1. A 1-page graphical syllabus of the material.  This is a diagram or flow chart that indicates how specific topics we covered fit into larger themes of the course.  We will brainstorm some themes in class.

2.  A 2-page chart containing major definitions and theorems from the class so far, with examples of the terms you defined and comments about why the theorems are important.  Think of this as a "cheat sheet" for an exam, but without the exam.  Please do not include everything; one purpose of this exercise is to decide what is important.  

30% Final Paper (Outcomes 5, 6)

See below.

Final Papers/Projects

In this course, you will undertake a major final project.  The goal of this project is to find some topic in topology or geometry that interests you, and understand it well.  You should aim to learn the equivalent of one week's worth of course material.  However, it will take much more than a week to research and learn one week's worth of course material!  In order to show me what you've learned, you will write a paper.  You may work in groups of up to three people, but keep in mind that I expect n times as much thinking from an n person group as I expect from a one person group.

The length of the paper is not as important as the quality of thinking, learning, and understanding reflected in the paper, but roughly, I expect it will take 5-10 pages for one person to demonstrate his/her understanding, 10-15 pages for two people to demonstrate their understanding, and 15-20 pages for three people to demonstrate their understanding.  

If you feel that the best way to express your understanding is by building a model, designing a lesson plan for a math workshop, writing a computer program, or really anything else (as long as I say it's okay), then you can do that, and write a shorter paper to go along with it.  No matter what, everyone must turn in some written text.  You'll certainly have references to cite, and you will want to explain what your model/program/etc. is doing.  If you write a program, it is your responsibility to meet with me in person and show me how to run it.  

Some time during the term, I will ask you to submit a short (<1 page) proposal for a topic.  In this proposal, you should include at least three references you will use to write your paper, and outline what you plan to learn.  The point of the proposal is to convince me and yourselves that you have picked a manageable topic, and to let me know if you plan to do anything in addition to writing a paper.  If you drastically change your plan at any point, it might be wise to show me a revised proposal.  The project should be fun, so if at some point you realize you are not having fun, I encourage you to find a different topic.

Student Needs

Students with disabilities enrolled in this course and who may need disability-related classroom accommodations are encouraged to make an appointment to see me before the end of the second week of the term. All discussions will remain confidential, although the Student Accessibility Services office may be consulted to discuss appropriate implementation of any accommodation requested.

I realize that some students may wish to take part in religious observances that fall during this academic term. Should you have a religious observance that conflicts with your participation in the course, please come speak with me before the end of the second week of the term to discuss appropriate accommodations.

Honor Code

On the homework:
I strongly encourage you to work on the homework problems in groups.  (Keep in mind that you will get the most out of the homework assignments if you are an active member of the group.  So come with ideas, even if you don't think they will work, and come with questions.)  You must write up your homework assignment independently--no copying is permitted.  In particular, your homework assignment must represent your own understanding of how to do the problems, and must be written in your own words.

On the quizzes:
Students may not receive assistance of any kind from any source (living, published, electronic, etc.), except the professor. 

On the portfolio and paper/project:
All submitted work must be your own (or work of group members in the case of the project), and all references must be properly cited.