Lyon College Associate Professor of Mathematics Dr. Jeremy Chapman and alumnus Benjamin Norton, ‘22, were recently published in Alpha Chi’s Journal of Undergraduate Scholarship. Their article proving that four noncoplanar points determine a unique sphere and finding an explicit formula for the center of a sphere can be found in Vol. 7, Issue 2, of the journal.

Chapman said, “The significance of our research is that spheres in three-dimensional vector spaces over finite fields are uniquely determined by four noncoplanar points, just as they are in the Euclidean setting.”

Chapman explained that the most significant difference in the two spaces is “the distinct vectors in the finite field setting can have zero distance from each other.” Something that caused the pair several issues that had to be contended with.

The article itself is the culmination of over two years' worth of work between the two. Norton began working on the project in the fall of 2020 where he spent most of his semester reading previous publications from Chapman to get a better understanding of the research.

“We started meeting regularly to discuss the research and individual progress made starting February 2021, leading up to the submission for the publication in April 2022,” said Norton.

Chapman added that a lot of the work had to be conducted over the summer due to the pair’s fall and spring schedules not meshing well.

The article is seven pages long, packed full of complex formulas, proofs, and references. Readers will also notice the addition of images taken from the popular game Minecraft. These images showcase graphs made using Minecraft’s building mechanics, which utilizes “blocks," to help visualize the article’s figures.

Norton, who was the mastermind behind using Minecraft, said, “Part of my role for this research was to create the visuals for the publication, as well as any graphics that I would need for presentations.”

Norton admitted he wasn’t terribly familiar with any of the programs he was using at the time, so he didn’t feel confident with creating something so intricate.

“Early on in the project when I was learning more about the proof and the objects we were using, I had built a Minecraft world to help represent the points since they were essentially just blocks,” said Norton. He continued, “Since I had already ‘drawn’ up some lines and spheres in that world while I was experimenting, I thought it would be neat to just take screenshots of what I had made.”

Surpassing “neat,” Norton said his Minecraft screenshots received great feedback.

“I’ve learned from conversation with friends and people who attended my presentations that the Minecraft examples made much more sense than either the math or the rudimentary graphics,” said Norton.

Chapman agrees that the use of the graphics did a good job capturing the three-dimensional aspects of the graph, and also “brought a fun vibe to the paper.”

In addition to the success of Minecraft’s added novelty to the article, Norton and Chapman said there were more nuances that they both enjoyed while creating the paper.

For Chapman, it was the experience of getting to conduct the research alongside a Lyon student.

“I have had several peer-reviewed publications in my career, but the two co-authored with Lyon students are extra special,” said Chapman.

For Norton, he said his proudest part might have to be the computations that make up the bulk of the third section of the proof.

“I scaled some of the previous work done in two dimensions up to three dimensions through the usage of linear algebra, which was not easy. It was very surreal to see how my handwritten calculations, covered in small notes and riddled with poor penmanship, were transformed into the very formal-looking work in the paper,” said Norton.

He continued, “I keep thinking it looks like it came out of a textbook.”

Norton ended with a word of advice to any future students thinking about getting into research.

“I think that any student who may be interested in doing research of any kind should not hesitate to ask their professors about the opportunities that the College can provide,” he said. “I was relatively late to the game for psychology, and for math I didn’t even know what my professors actually studied.”

He continued, “Doing research with faculty members is definitely an experience I would love for other students to experience as you get to learn more about the discipline and you are given the opportunity to build stronger relationships with your professors.”

If you are interested in learning more about the pair’s publication, you can view it here: https://alphachihonor.org/headquarters/files/Website%20Files/Aletheia/Volume_7_2/AX0344_Norton.pdf