An undergraduate seminar during the spring of 2022.
Tilings, sometimes known as tessellations, abound: look to nature, architecture, art, and video games. There are much more exotic tilings which, for example, are aperiodic.
Tilings are pretty. Tilings also naturally lead to many mathematically interesting questions. For example:
And so forth! This seminar will touch on some of this vast topic of tilings.
Each participant will give 1 or 2 talks over the course of the semester, during which I hope one enjoys some interesting mathematics, as well as meditates on the art of presenting and talking about mathematics. During the rest of the semester, I hope that those that are not presenting will help form a lively and friendly seminar environment, one in which all participants can feel comfortable to discuss and learn in.
There are many possible topics in the mathematics of tilings to explore. What follows is a sampling of possible topics and references for this seminar. This list is by no means exhaustive, and I would encourage you to find something that suits your taste!
This is a quick survey, aimed at a general audience, on the mathematics of tilings. The text contains many references for further reading.
This is perhaps the first textbook on the mathematics of tilings. It is full of pictures and gives a rather comprehensive treatment of many topics around tilings of the plane. Problems about tiling the plane are still topics of current research!
This book is a more recent treatment about tilings, especially in regards to the symmetries they admit. The first part of this book gives a pretty and gentle classification of the possible symmetry groups of plane tilings—otherwise known as crystallographic groups. The later chapters explore more advanced topics involving group theory and hyperbolic tilings. It’s a book full of pictures and ideas, and should be an absolute delight to read.
Have you played Tetris? Or perhaps you have packed a grocery bag. In any case, this is a game of tiling by tetromino. A polynomino is like tetris-like blocks of different sizes. You can now ask many questions about them, and this book is a guide to some of the mathematics.
Aperiodic tilings are ones which do not admit translational symmetry. Before reading anything, you can do a lot worse than taking a look at the following gallery of substitution tilings:
One of the most well-known aperiodic tilings are the Penrose tilings. Some things to read around this include:
Roger Penrose (1974), The role of aesthetics in pure and applied mathematical research.
Roger Penrose (1979), Pentaplexity: A Class of Non-Periodic Tilings of the Plane.
Since then, there are many construction methods for aperiodic tilings. Some other interesting resources include:
Robert Berger (1966), The undecidability of the domino problem.
Charles Radin (1994), The pinwheel tilings of the plane.
Charles Radin (2021), Conway and Aperiodic Tilings.
John Horton Conway invented a fascinating group-theoretic method to prove whether or not a region admits a tiling. Some basic references for this are the papers:
John H. Conway and Jeffrey C. Lagrais (1990), Tiling with polynominoes and combinatorial group theory.
William P. Thurston (1989), Groups, tilings, and finite state automata.
William P. Thurston (1990), Conway’s Tiling Groups.
Igor Pak (2000), Tile Invariants: New Horizons.
The Aztec diamond is like a pixellated diamond, see this picture. Many questions about domino tilings of Aztec diamonds admit pretty solutions. Some references on this topic might be:
Noam Elkies, Greg Kuperberg, Micahel Larsen, and James Propp (1992), Alternating-Sign Matrices and Domino Tilings (Part I).
Noam Elkies, Greg Kuperberg, Micahel Larsen, and James Propp (1992), Alternating-Sign Matrices and Domino Tilings (Part II).
William Jockusch, James Propp, and Peter Shor (1998), Random Domino Tilings and the Arctic Circle Theorem.
We meet on Fridays from 1 to 3 PM in Mathematics Room 507.
