Math and research aren’t two words that seem to fit together quite as well as science and research. Indeed, math research looks much different than physical or life sciences research—where one can easily picture white lab coats, petri dishes and microscopes.
Instead math researchers work with theorems, axioms and definitions in their mathematical proofs. This summer, Ryan Childress ’26 and Emma Heisig ’28 used spheres to conduct math research in discrete and convex geometry as part of Longwood’s undergraduate research program, PRISM.
I hope students get a new appreciation for math—both what it actually entails and what it means to be a mathematician or a scientist.
Dr. Steven Hoehner, associate professor of mathematics Tweet This
“Math research is proving new theory. So, proving new things that are known or giving new proofs of things that are known. We are proving new results,” said Dr. Steven Hoehner, associate professor of mathematics, who worked one-on-one with Childress and Heisig on their project. “I hope students get a new appreciation for math—both what it actually entails and what it means to be a mathematician or a scientist.”
This is Hoehner’s fifth time serving as a research mentor in the Perspectives on Research in Science and Mathematics (PRISM) program, an eight-week summer research program that pairs a faculty member with one or two students to work collaboratively on a project in biology, chemistry, environmental science, mathematics, physics and psychology.
Childress is a math major concentrating in secondary education, with plans to teach after graduation. Heisig is a physics major with an engineering concentration and a math minor. We caught up with both students to learn more about their experience in PRISM.
Tell us about your project.
Childress: We’re working with a unit sphere (centered at the origin (0,0,0)). The goal is to place five spherical caps, or balls, on the surface of the sphere. As their radius increases, we want to find the best way to arrange them to partially cover the sphere. We are not trying to pack them tightly and we’re not trying to cover the whole sphere. Instead, we’re somewhere in between—looking for the most optimal coverage, possibly with some overlap.
Heisig: It gets into some pretty complex geometry. A unit sphere might seem simple—it’s the same distance from the center in every direction—but once you start placing points on it, things get tricky. You’re working with different coordinates, and a lot of unexpected constraints come up. That’s especially true when placing five points, which is much more complicated than just placing two.
What does a typical day of research look like?
Childress: I think of our work as we’re kind of a think tank. We work collaboratively, just kind of bouncing ideas off of each other. Of course, Dr. Hoehner being the research mentor, he’s pushing us in the right direction, and definitely doing some of the uplifting.
Heisig: Most people think of laboratory research when you hear the term researchers. With that type of research you’re doing maybe 100 trials of a specific process, and you know you’re going to get a result, but you don’t know if that result is going to help you or hurt you. With this, it’s a bit more fluid. You know you’re heading down a path in a direction, and that may take you all the way to the finish line, or you may get halfway there and realize you have to go back and take a completely different way. It’s kind of start and stop, but it’s pretty fun.
What are some applications of this research?
Heisig: This has some really cool and important applications. A big one is satellites, which are very relevant today. It also shows up in things like error-correcting codes and even crystallography—looking at molecules on a tiny scale. The paper we’re using to base some of our work on and referencing is actually by a chemist, so it’s not just math—it’s a mix of fields coming together.
I think of our work as we’re kind of a think tank. We work collaboratively, just kind of bouncing ideas off of each other.
Ryan Childress ’26
What does a typical day of research look like?
Childress: I think of our work as we’re kind of a think tank. We work collaboratively, just kind of bouncing ideas off of each other. Of course, Dr. Hoehner being the research mentor, he’s pushing us in the right direction, and definitely doing some of the uplifting.
Heisig: Most people think of laboratory research when you hear the term researchers. With that type of research you’re doing maybe 100 trials of a specific process, and you know you’re going to get a result, but you don’t know if that result is going to help you or hurt you. With this, it’s a bit more fluid. You know you’re heading down a path in a direction, and that may take you all the way to the finish line, or you may get halfway there and realize you have to go back and take a completely different way. It’s kind of start and stop, but it’s pretty fun.
I think research requires using different muscles, for lack of a better term, than taking a class during the academic year. Especially with it being focused on one project, instead of five different classes. It’s like a marathon, but yet, really rewarding.
Emma Heisig ’28 Tweet This
What has been your favorite part of research so far?
Childress: I’ve been working with a lot of math coding software, and that’s been fun trying to get it to do exactly what I need it to do. Sometimes you break the algorithm, sometimes you make the algorithm. Problem solving is fun!
Heisig: I think research requires using different muscles, for lack of a better term, than taking a class during the academic year. Especially with it being focused on one project, instead of five different classes. It’s like a marathon, but yet, really rewarding.
Would you recommend PRISM to other students?
Childress: I’d recommend it to anyone who is interested in learning more and going beyond their major. It’s cool because you are contributing to the wealth of knowledge of a discipline. I would recommend it to anyone who is interested in actually seeing how these things are discovered or invented.
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