# Mathematics Research

## Mathematics Research

##### Dr. Mike McDaniel has written a book, *Geometry by Construction: Object Creation and Problem-solving in Euclidean and Non-Euclidean
Geometries*, which uses the results of his many successful undergraduate research projects at
Aquinas. Read more.

### May 2024 - May 2025

**Using VR to find conjectures in spherical geometry**

**Michael Vanderkolk**

By modeling geometric ideas in spherical geometry using Blender to make meshes, animate and render, we have already found new conjectures. Conjecture 1: the set of twelve projection points and Wallace-Simson lines published previously occurs for the most general case of one single projection point with a Wallace-Simson line. Conjecture 2: A subset of Wallace-Simson lines and lines of projections and feet of projections and points of projection form a natural 12-point, 12-line Projective Geometry. (This would be news because the 12 point Projective Geometry has been proved to exist but no actual example of one has been found.) Conjecture 3: A subset of projection points, feet of projections, etc forms a 7 point Projective Geometry isomorphic to the Fano Plane. Our Blender examples have already bloomed into very interesting spherical models. One modification uses collars, like Saturnian rings, which stick out from the sphere for spherical great circles. We can modify how far the collars stick out to emphasize importance. Since we intended to fit as much information into the mesh as we could, having a new way to see our ideas, a way completely impossible in 2-D, shows immediate promise. Next we will work on creating animations to illustrate the complicated ideas we are exploring.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics ****Funded by: Mohler-Thompson Summer Research Grant**

### May 2023 - May 2024

**Efficient Movement with the Mancala Mechanism ****Edin Mehanovic**

We will explore various problems involving the Mancala mechanism. These will include finding bounds for efficient movement of stones and variations on the puzzle Tchoukaillon.

**Faculty Advisor: Dr. Joseph Spencer, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

**Properties of elliptic Wallace-Simson lines and applications in three dimensions.****Rebecca Beltran**

All the elliptic geometry developments done through Mohler-Thompson since 2018 have applications in three dimensions. We plan to model with VR in Blender to create animations to illustrate our ideas. We also plan to develop double and triple integral formulas based on our geometry.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

### May 2022 - May 2023

**Evaluating Secondary Mathematics Teacher Preparation at Aquinas CollegeNicole Gregory**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 Teacher 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.

**Faculty Advisor: Dr. Shari McCarty, Mathematics**

**Funded by: Mohler-Thompson Summer Research Grant****Using Industry Insights to Create a Project-Based Approach to Data Analytics**

**Ronnie McMillan**

We will conduct a series of interviews with data science practitioners, specifically those who regularly use the statistical programming language R, to gain insights about the skills that are considered valuable in the professional data analytics community. We will use these insights to rework the course MS 282 (Applied Statistics with R), which is a foundational course in the Data Analytics major, making it into a completely project-based experience. This revamping will consist of writing and testing a series of projects based on industry-provided datasets.

**Faculty Advisor: Dr. Joe Fox, Mathematics**

**Funded by: Aquinas College Summer Scholars Program**

**A Plethora of Wallace-Simson LinesJarrad Epkey**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.

**Faculty Advisor: Dr. Michael McDaniel, Mathemathics**

**Funded by: Mohler-Thompson Summer Research Grant**

### May 2021 - May 2022

**Computing Group Representation Using SAGEShekira Edgar**

A group is a mathematical structure that encodes symmetry, and a representation of a group is a way to realize what those symmetries are using the language and techniques of linear algebra. This project will analyze the representations of certain groups known as the classical groups, which are groups of matrices. Our problem can be approached from an elementary point of view, and most of our work will be devoted to automating the necessary calculations by writing short computer scripts.

**Faculty Advisor: Dr. Joe Fox, Mathematics ****Funded by: Mohler-Thompson Summer Research Grant**

**Elliptic Geometry Applied to Biochemistry and Geospatial AnalysisNoelle Kaminski**

We are building on the last three years of Mohler-Thompson summer research. We have new math results which could have consequences beyond geometry. For instance, the icosahedral viral capsid has projection lines with collinear feet in important places. Also, satellite orbits fit elliptic geometry very nicely. Can we say anything new? We're going to try.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics ****Funded by: Mohler-Thompson Summer Research Grant**

### May 2020 - May 2021

**Elliptic geometry applied to icosahedral capsids.Joshua Wierenga**

Dr. David Wilson's 2016 result that, for viruses with icosahedral shapes, viral probes lie on icosahedral great circles inspires us to seek further structures through geometry and bio-chemistry. We will study his paper and its references along with articles on the geometry of the icosahedral sphere. We plan to build physical models using Zometool and virtual models in SketchPad. We aim to find a few properties which are new to Dr. Wilson and send our findings to him for verification.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

**Tchoukaillon CrossingsThomas Siebelink**

Tchoukaillon is a solitaire game that uses the sowing mechanism from game Mancala. Tchoukaillon’s board consists of a single row of bins. The goal is to sow all stones off the board to an end bin, called the Ruma, with the requirement that each sow ends in the Ruma. It is known for what arrangements of stones that Tchoukaillon can be won using a fairly simple strategy. We will study winning strategies when there are multiple rows to sow from and the rows cross at some bins or bins.

**Faculty Advisor: Dr. Joseph Spencer, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

### May 2019 - May 2020

**Monsky's Theorem and the Wallace-Simson Line in Non-Euclidean GeometryMorgan Nissen**

We will seek other counter-examples to Monsky's Theorem in hyperbolic geometry and we will seek the set of points from which an elliptic triangle has Wallace-Simson lines.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

**The Mathematics of Machine Learning AlgorithmsAnna Putnam**

We are creating, from scratch, an image recognition program using neural networks, basic programming, and calculus concepts. Our goal is to have a working program that can label images of oak leaves and maple leaves accurately, through exploring the mathematics of machine learning.

**Faculty Advisor: Dr. Joe Fox, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

### May 2018 - May 2019

**Acyclic orderings in citation networksAimee Judd**

Citation networks are large collections of objects, some of which refer to others. For example, one might consider a collection of Supreme Court decisions, some of which cite others as precedents. In this project, we will develop a method to count all possible orderings of the objects in a citation network that preserve the citation structure.

**Faculity Advisor: Dr. Joe Fox, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

**Monsky's Theorem and Pedal Triangles in Non-Euclidean GeometryKelsey Hall**

We will calculate examples to show how Monsky's Theorem fails to be true in non-Euclidean geometry. We will seek theorems for pedal triangles in non-Euclidean geometry.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics****Fund by: Mohler-Thompson Summer Research Grant**

### May 2017 - May 2018

** Efficient Mancala MovementHolly Ensley**

This project investigates the mechanism of movement used in the game mancala. We are interested in how we can most efficiently move particular stones, or kings, over long distances. We will explore cases with different numbers of kings and starting stones. Through this, we are hoping to find a bounds for measurements of efficient movement.

**Faculty Advisor: Dr. Joseph Spencer, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

**Hyperbolic versions of Euclidean theoremsPaul Gass**

Napoleon, Pascal, Brianchon, Miquel, and many other geometers have theorems bearing their names. Some of these theorems hold in hyperbolic geometry, some fail and some require further investigation. We hope to add a few new hyperbolic versions of these named theorems.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

### May 2016 - May 2017

** Hyperbolic QuadraturesTristen Spencer**

Previous M-T researchers have squared the circle in non-Euclidean spaces. This opens
the door to explore other hyperbolic cases of areas of circular regions that match
areas of polygonal regions. We hope to show the five Euclidean quadratures fail and
discover new hyperbolic quadratures which would be impossible in Euclidean space.**Faculty Advisor: Dr. Michael McDaniel, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

** Investigating Movement in MancalaMaria Maguire**

This project investigates the mechanism of movement used in the game mancala. We are interested in how we can most efficiently move particular stones, or kings, over long distances. We will explore cases with different numbers of kings and starting stones. Through this, we are hoping to find a pattern of most efficient movement.

**Faculty Advisor: Dr. Joseph Spencer, Mathematics****Funded by: Mohler-Thompson Summer Research Grant**

### May 2015 - May 2016

**Jacob Campbell**

Jacob Campbell and Dr. Fox researched the properties of circulant graphs, including their independence numbers.

**Faculty Advisor: Dr. Joe Fox, Mathematics****Cecilia Magnuson**

Cecilia Magnuson and Dr. McCarty researched higher-order thinking in middle school mathematics classrooms.

**Faculty Advisor: Dr. McCarty, Mathematics**### May 2014 - May 2015

**Krystin Dreyer**

Krystin Dreyer and Dr. Fox studied the mathematics of citation networks.

**Faculty Advisor: Dr. Joe Fox, Mathematics****Kyle Jansens**

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.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics**### May 2013 - May 2014

**Noah Armstrong**

Noah Armstrong and Dr. Spencer studied the mathematics of the game Mancala.

**Faculty Advisor: Dr. Joseph Spencer, Mathematics****Noah Davis**

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.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics**

May 2012 - May 2013

**Jackie Gipe**

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.

**Faculty Advisor: Dr. Michael McDaniel, Mathematics***Rose-Hulman Undergraduate Math Journal*, 2013. Read it here.

*Pi Mu Epsilon Journal*, Spring 2011, Vol. 13, No. 4. Read it here.

*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.