
Math 4096: Geometry of Tilings / Senior Problem SolvingFall Semester 2021
Topic and bookTextbook: 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 structureIn 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 WritingIntensive 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:
Grading scheme and policies
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:
Here are a few guidelines for how to write up the proofs:
Final exam policy: The final exam will only be administered during the stated final exam period: Thursday, December 9, from 1:003:00pm. A student who misses the final exam will fail the course. Exceptions will only be granted in the most grave and welldocumented of circumstances.
Other policiesAcademic 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. dfuter at temple edu Last modified: Mon Aug 29 13:41:22 EDT 2016 