Mathematics at Aquinas College
Undergraduate Research Projects
Dr. Mike McDaniel has written a book, Geometry by Construction: Object Creation and Problem-solving in Euclidean and Non-Euclidean Geometries, which uses results of his many successful undergraduate research projects at Aquinas. Read more.
Current student research
Evaluating Secondary Mathematics Teacher Preparation at Aquinas College
Nicole Gregory and Dr. Shari McCarty
The project will evaluate what it means to mesh content knowledge and pedagogical content knowledge together within a traditional mathematics major at Aquinas College. To thoroughly prepare future teachers, we aim to keep in mind the mathematical content (general mathematics program), the pedagogical content (courses within the School of Education), and how to develop touch-points for pedagogical content knowledge. All experiences need to blend together to fully prepare future teachers for the Michigan Test for Teachers Certification (MTTC) and overall readiness to teach in the classroom. Through the lens of how to best prepare future teachers for their classrooms, this project will design teacher-preparation specific language for traditional math major course descriptions and syllabi, design multiple experiential learning opportunities in clinical settings, and design opportunities for assessment of key skills throughout the program.
A Plethora of Wallace-Simson Lines
Jarrad Epkey and Dr. Mike McDaniel
The Wallace-Simson theorem in elliptic geometry is completely different from the Euclidean version, which was completely defined in 1797. Building on previous summer work, we are finding many projection points for Wallace-Simson lines. We can prove some are non-constructible. We hope to find ways to count and construct projection points and to see how these projection points and Wallace-Simson lines interact with other properties of elliptic triangles.
Other recent student research:
- Shekira Edgar and Dr. Joe Fox worked on a project in representation theory, writing a computer program using Python to work out the structure of certain representations of the classical groups.
- Noelle Kaminski and Dr. Mike McDaniel investigated applications of elliptic geometry in viral capsids, discovering new ways to measure distances and new properties of projections.
- Thomas Siebelink and Dr. Joe Spencer developed formulas and algorithms that can be used to create winning strategies in the game Tchoukaitlon.
- Josh Wierenga and Dr. Mike McDaniel created models of icosahedral viruses using structures from elliptic geometry. Their project led to a paper which has been accepted for publication in Mathematics Magazine.
- Morgan Nissen and Dr. Mike McDaniel extended the work begun by Dr. Kelsey Hall in 2018-19, exploring properties of Wallace-Simson lines of triangles in elliptic geometry. Morgan, Kelsey, and Dr. McDaniel's work in this area is currently under review for publication.
- Anna Putnam and Dr. Joe Fox researched the mathematics of image recognition algorithms, writing a program that uses a neural network to identify tree species from a picture of its leaves.
- Kelsey Hall and Dr. Mike McDaniel proved a Wallace-Simson-type theorem in elliptic geometry and gave some counterexamples to Monsky's theorem in hyperbolic geometry.
- Aimee Judd and Dr. Joe Fox investigated properties of directed acyclic graphs. Their work was published in the peer-reviewed journal Ars Combinatoria in 2020.
- Holly Ensley and Dr. Spencer continued work from the previous year, studying mathematical problems arising from the game Mancala.
- Paul Gass and Dr. McDaniel extended results from Euclidean geometry to hyperbolic geometry.
- Maria Maguire and Dr. Spencer initiated a study of certain mathematical problems arising in the game Mancala.
- Tristen Spencer and Dr. McDaniel completed a project, “Euclidean measurements for hyperbolic constructions”. They have submitted a paper for publication which is currently under review.
- Jacob Campbell and Dr. Fox researched properties of circulant graphs, including their independence numbers.
- Cecilia Magnuson and Dr. McCarty researched higher order thinking in middle school mathematics classrooms.
- Krystin Dreyer and Dr. Fox studied the mathematics of citation networks.
- Kyle Jansens and Dr. McDaniel completed the long story of squaring the circle. It’s been known since 1882 that in Euclidean geometry, it’s impossible to construct, using only a straightedge and compass, a square with the same area as a given circle. In 2012, Noah Davis and Dr. McDaniel proved that such a construction is possible in hyperbolic geometry, and Kyle and Dr. McDaniel proved it’s also possible in elliptic geometry. Their result was published in the Rose-Hulman Undergraduate Math Journal, and Kyle, co-author Noah Davis, and Dr. McDaniel even made the local news! Read Kyle’s and Noah’s paper here.
- Noah Armstrong and Dr. Spencer studied the mathematics of the game Mancala.
- Noah Davis and Dr. McDaniel proved that it’s possible to square the circle in hyperbolic geometry. Noah presented his results at MathFest in Portland, Oregon, and his paper was published in Rose-Hulman Undergraduate Math Journal. Read Noah’s paper here.
- Jackie Gipe and Dr. McDaniel worked on a project titled, “Invertible Chord Diagrams from the Wheel.” Jackie presented her work at a Calvin College colloquium.
- In addition to the publications mentioned above, AQ math student publications include:
- “Tangent circles in the hyperbolic disk”, by Megan Ternes. Appeared in Rose-Hulman Undergraduate Math Journal, 2013. Read it here.
- “Fibonacci numbers and chord diagrams”, by Jane Kraemer. Appeared in Pi Mu Epsilon Journal, Spring 2011, Vol. 13, No. 4. Read it here.
- “Hyperbolic polygonal spirals”, by Jillian (Russo) Duffy. Appeared in Rose-Hulman Undergraduate Math Journal, Fall 2010 Vol. Read it here.
- “Alhazen’s hyperbolic billiard problem”, by Nate Poirier. Appeared in Involve, Vol. 5, No. 3, 2012. Read it here.