Tile Invariants and an Exploration of Tilings with Ribbon Pentominoes and L-Pentominoes

Lucy Wickham

Name: Lucy Wickham
Major: Mathematics
Advisors: Subhadip Chowdhury, Pamela Pierce

In this Independent Study, we survey the mathematics of tiling 2-dimensional regions with polyomino shapes of varying sizes. We investigate tile invariants to prove tileability and examine specific tile invariants, such as the Conway/Lagarias invariant. Using “Tile Invariants for Tackling Tiling Questions” by Dr. Michael Hitchman as a guide for exploration, we survey different techniques for finding tile invariants, such as coloring, boundary words, height, and group theoretic techniques. After this background is established, we answer an open problem posed by Hitchman in the affirmative – we prove the requirements for a modified rectangle to be tileable by area 5 ribbon tiles. In the final part of this project, we consider L-pentominoes and conjecture the requirements for a rectangle to be tileable by this tile set. We prove the conjecture in certain cases.
This project is interesting to me because it broadened my knowledge about the applications of group theory and is an interesting example of the intersection of that area of mathematics with the topic of combinatorics. I was excited to pursue this project because it has taught my invaluable skills about proof writing, problem solving, and new areas of combinatorics, which will help me to continue researching theoretical mathematics in graduate school.

Posted in Comments Enabled, Independent Study, Symposium 2023 on April 14, 2023.

9 responses to “Tile Invariants and an Exploration of Tilings with Ribbon Pentominoes and L-Pentominoes”

  1. Josh & Eli says:

    Super cool project…. and thanks for all the help in office hours!!!

  2. Prof. Claire Eager says:

    Congratulations, Lucy! It’s been a long time since I thought about combinatorics and tessellation (though the latter came up in my Green Shade: Poems in Place class’ visit to the art museum yesterday). I love how clear and elegant the poster is for a non-expert, and I now have Square 1 TV and/or Tetris music stuck in my head . . .

  3. Barnaby says:

    That’s an impressively dense poster Lucy! I’m curious to know how much of this project was researching existing knowledge, versus conducting your own work to advance understanding of this topic?

  4. Emily Sippel says:

    Congrats, Lucy! Writing 80 pages about anything is impressive 🙂

  5. Lucy Wickham says:

    The background research is all existing knowledge, and then I build upon that body of mathematics in the two tiling problems I tackled. In the “Tiling Modified Rectangles with Ribbon Tiles” section of the poster, the results that were my own are Theorem 3.4 and Lemma 3.1 and 3.2. In the “Tiling Rectangles with L-Pentominoes” section, all the theorems and lemmas are new results that I proved myself.

  6. Kate Wickham says:

    I’m off to play tetris. Great poster and research

  7. Mark Graham says:

    Hi Lucy — congratulations on completing your IS, and thank you for sharing it here!

  8. Jen Druliner says:

    This poster is great, Lucy! Your writing is clear and your math skills are, in theory, great (see what I tried to do there?). As a history person, I am in awe of anyone who can do what you did in this research. You are amazing!

  9. kay campbell says:

    In other words, your findings uncovered some new math! One person can really make a difference. Thank you!