Emma Miller '22

2020 SOAR Profile

The Clumsy Packing of Polyominoes and the Strong Proper Connection of Graphs

Major and Minor: Math and Physics
Hometown: Drums, PA
Project Advisor(s): Dr. Nathan Shank

Briefly describe your project.

Over the course of the summer, I worked on two different projects in collaboration with the National Science Foundation Research Experience for Undergraduates (REU) program in Computation and Experimental Mathematics (CEM) at ɫ��ɫ. The one project I worked on was pertaining to the clumsy packing of polyominoes and was led by Dr. Shank. My other project was about the strong proper connection of graphs and was led by Dr. Caitlin Owens from DeSales University and Dr. James Hammer from Cedar Crest College.

For my project with Dr. Shank, we focused on polyominoes, which are sets of squares, or more formally cells, such as the pieces seen in Tetris. The number of cells in a polyomino is defined as n. We then looked at packing these polyominoes clumsily on an n x n grid. A clumsy packing is the minimum number of polyominoes that can be arranged in a grid such that no other polyominoes of the same size can also be placed in that grid. In addition to the clumsy packing of a polyomino, we also looked at the uniqueness which refers to special cases where only one packing of a specific size exists for a specific polyomino in a specifically sized grid.

The second project I worked on was pertaining to the strong proper connection of a graph. A graph is a set of vertices and edges. We looked at different classes of graphs as well as a variety of graph operations. The strong proper connection of a graph is a coloring of some graph such that there exists a properly edge colored shortest path between any two vertices within the graph. A path is a sequence of vertices such that each vertex shares an edge to the next vertex while a properly edge colored path has no two adjacent edges with the same colors.

Describe the origin of your project. (E.g., did you pitch the idea and choose a faculty member, or did they come to you with an idea?)

Dr. Shank recommended that I apply for a SOAR project so that I could work in collaboration with the REU and get some research experience under my belt. That being said, we didn’t have a concrete idea of what I was going to work on—only a vague concept based on the projects already out on the REU website. Once the REU and my SOAR project started, I was able to pick two of the projects from the long list of REU-supported options.

What’s the best part about working with your faculty mentor? What valuable insights have they brought to your project?

The best part about working with Dr. Shank was actually being able to work with him on the project. Instead of me being in a lab and doing the project and problems independently, Dr. Shank, as well as my other mentors and group members, were all hands on deck while approaching the problem. While we all approached the problems from different angles and different thoughts for the solution, we all came together to do some great work. Dr. Shank, along with the other REU mentors, provided valuable insight into the inner workings of a research project in mathematics, which are a lot different compared to other disciplines, such as physics. In addition, they also provided insights into how to pursue research as a future career and what research in graduate school looks like.

What has been your biggest obstacle so far?

One of the biggest obstacles so far is overcoming the barriers of hard problems. Some theorems we were able to prove in just a day, but other ones we are still trying to figure out. For some of the problems, such as the free rectangular polyominoes, it seemed like we kept getting the wrong answer after the wrong answer, which became very discouraging. To get over this obstacle, we would approach different problems in hope that things we learn from working with other polyominoes or graphs would aid in the harder questions—and sometimes they did, especially in the strong proper connection group.

What has been your biggest takeaway from this experience?

The biggest takeaway from this experience is how to work with a group from all different backgrounds on a single problem, or problems. Even though I was familiar with math research prior to this project, the people I was working with had taken the same courses and had the same background in mathematics that I did. Being able to work with not only experienced professors but also other students from a number of different schools with a number of different backgrounds was very interesting. It showed me the importance of realizing people’s strengths and weaknesses and how to work efficiently in a group.

What was the result of your project?

The polyominoes group I worked with had proven roughly 20 theorems pertaining to the clumsy packing of polyominoes on a finite grid, which was not widely studied previously. Similarly, the strong proper connection group, which was notably double the size of the polyominoes group, had proven roughly 40 theorems pertaining to the strong proper connection of graphs. These things that we had proven were not minuscule things either—for example, in the strong proper connection group, we were able to prove conjectures and open questions that we found other people who researched this project previously were not able to prove or even have an idea of how to prove.

In your own words, how do you feel about being awarded this opportunity? Why should other students take advantage of the SOAR program at ɫ��ɫ College?

I am extremely grateful for being able to be awarded this opportunity. The ability to complete summer research has really opened my eyes and opened opportunities to the world of math research and how I can pursue it as a future career. I think other students should take advantage of the SOAR program at ɫ��ɫ because it provides really good insight into what research actually looks like outside of a classroom setting.

Now that SOAR is over, do you plan to expand upon your research? If so, how?

Both groups hope to continue their research and hopefully publish a paper by the end of the year. In addition, I hope to extend some of the research I did here to other projects in the future.

Have you, or do you plan to present this research outside the SOAR presentations? If so, where? Be specific, if possible.

As it stands now, due to the availability of my groupmates and the current world climate, we do not have plans to present this research outside the SOAR presentations in the foreseeable future. However, we hope to eventually.