Math 4096: Geometry of Tilings / Senior Problem Solving

Fall Semester 2021

Meets: Tue/Thu 3:30-4:50 in Wachman Hall, room 010
Canvas page: https://templeu.instructure.com/courses/98240
Instructor: David Futer
Office: 1026 Wachman Hall
Office Hours: Tue 11:00-12:00, Tue 1:30-2:30, Thu 11:00-12:30, and by appointment
E-mail: dfuter at temple.edu
Phone: (215) 204-7854

Topic and book

Textbook: The Tiling Book, by Colin Adams. The book is a PDF preprint, available from the course Canvas page. The author has generously shared the text with this class, for free. At the author's request, please do not post it anywhere on the public web.

Prerequisites: Some background in Real Analysis and Modern Algebra. Math 3096 or 3098 would be particularly useful. Some experience with writing proofs.

Course topics: We will study the geometry and symmetry of tilings. After developing an appropriate language to describe tilings and the patterns that generate them, we will learn to describe their symmetries in the language of group theory. We will then prove that there are exactly 17 different groups that arise as symmetries of tilings of the plane. (Why 17? You will have to stick around to find out!)

In the second half of the semester, we will study aperiodic tilings - those with no symmetry at all - as well as the types of tilings that arise in curved geometry.

Why tilings? The theme of Math 4096 changes every semester. Why did I choose tilings?

I find the subject to be beautiful and visual. It has a rich history: elaborate tilings have been studied in virtually every human civilization. The subject marries geometric intuition with algebraic rigor. Finally, this subject has a "choose your own adventure" feel that leaves lots of room for individual exploration.

Course structure

This course will differ from most math classes. The rough structure of a class day will be 50% lecture from the instructor and 50% inquiry-based learning.

In particular, students will spend significant class time working together in groups of 3 or 4 trying to work out important examples and prove results that are central to developing the theory. The goal is to learn mathematics by doing mathematics rather than listening to a lecture with the prepackaged correct answer. More specifically, the point is not only to know what is true, but to discover why. You will learn the material the way a research mathematician solves a problem: through experimentation, trial and error, discussion with your peers, and a lot of hard work. Another key component of this is presenting your work to your peers.

A Writing-Intensive Course Math 4096 has been designated a "writing in the disciplines" course. This means we will devote a lot of attention to writing (and also oral communication). This will be visible in a few ways:

  • Classwork. During inquiry-based blocks of class time, we will practice working out and writing proofs of important results. You will also be asked to explain the results and ideas to your classmates. To encourage trial-and-error exploration, classwork will not be graded for correctness. (However, actively participating in these activities will be the key component of the participation score.)
  • Proofs in the homework. Homework assignments will mostly consist of proofs. They will be graded both on correctness and on clarity of exposition.
  • In-class presentation. At the end of the semester, every student will give a 15-20 minute presentation on a topic of their choice, chosen from a menu of options. Two weeks before the presentation, you will need to turn in a detailed outline, for discussion and revision. You are likely to find that presenting difficult mathematics orally is just as critical as presenting it in a written form, and involves somewhat different emphases.

Grading scheme and policies

Component Date Worth
Class participation Daily 30%
Homework Most Thursdays 30%
Presentation After Thanksgiving 20%
Final Exam December 9 20%

Attendance policy: Participation is fundamental in this course, hence attendance is mandatory. Absences will only be excused by documentation from a doctor or other health professional, Student Health Services, other University office, etc. Every unexcused absence will lower your class participation score by 1/3 of a letter grade (A- to B+, B+ to B, etc).

Homework policy Homework assignments will be posted on Canvas, and will typically be due on Thursdays. No late homework will be accepted, but I will drop your lowest homework score. I encourage you to start early and work in groups. There are only a couple of caveats to group work:

  • You should try to do all of the problems on your own before getting together with others. It does not benefit you (on exams and in the real world when you need to use math) to simply get solutions from your classmates! In fact, there is research suggesting that group work is much more productive when everyone has thought about the problems before getting together.
  • Everyone must turn in their own solutions. In other words, all written work must be your own, and written in your own words.

Here are a few guidelines for how to write up the proofs:

  • Write up the problems in order, using only one side of the page and leaving lots of space for comments. Please staple your sheets together.
  • Begin each problem with a statement of that problem.
  • Proofs should be written in complete sentences, with appropriate mathematical notation where appropriate. Pictures can certainly be used to illustrate the text, but the text still needs to make sense.
  • Proofread what you've done to be sure that it's complete and makes sense. Remember that proof-writing is above all an act of communication, and that the ultimate goal is clarity.
  • Start early! This way, if you are stuck, you can still discuss the problem with the other students, or with me.

Final exam policy: The final exam will only be administered during the stated final exam period: Thursday, December 9, from 1:00-3:00pm. A student who misses the final exam will fail the course. Exceptions will only be granted in the most grave and well-documented of circumstances.

Other policies

Disability statement: Any student who has a need for accommodation based on the impact of a documented disability, including special accommodations for access to technology resources and electronic instructional materials required for the course, should contact me privately to discuss the specific situation by the end of the second week of classes or as soon as practical. If you have not done so already, please contact Disability Resources and Services (DRS) at drs@temple.edu or 215-204-1280, or in person at 100 Ritter Annex, to learn more about the resources available to you. I will work with DRS to coordinate reasonable accommodations for all students with documented disabilities.

Academic rights and responsibilities: Freedom to teach and freedom to learn are inseparable facets of academic freedom. The University has a policy on Student and Faculty and Academic Rights and Responsibilities (Policy #03.70.02)

Academic Honesty Statement: Please see here.

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Last modified: Mon Aug 29 13:41:22 EDT 2016