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Name: Andrew William Watson ’12
Honors in: Mathematics
Hometown: Whitehall, Pa.
Major: Mathematics and Physics
Title of project: On the Unfolding of Prisms
Project advisor: Dr. Kevin Hartshorn
Abstract or brief description: Geometry is one of the most widely studied subjects in human history. Plane figures, called polygons, have been sketched and investigated for millennia. And polygons have three-dimensional analogs called polyhedra, which have been the subject of much inquiry since at least the time of the Greek philosophers. More recently (since about the 16th Century C.E.), the idea that polyhedra can be “unfolded” has been under investigation by mathematicians. These unfoldings, called nets, have become the standard expression of a three-dimensional polyhedron in two-dimensional Euclidean space. However, whether or not certain classes of polyhedra can be expressed as nets, which do not overlap themselves, is currently unknown, and this is where this work comes in.
It has been proven that some classes of polyhedra always have such a net. And conversely, for some other types of polyhedral, it has been proven that no such nets exist. Some polyhedra which are not edge-unfoldable, however, are vertex-unfoldable or generally unfoldable, two less-strict requirements for “unfoldability.” The goal of this project is to identify whether a few more classes of polyhedra can be unfolded through one of these methods. My work shows that all genus-0 prisms, convex or not, can be unfolded given certain conditions. I've also shown that certain prisms with genus can be unfolded given certain conditions, or made to be generally unfoldable via a few distinct methods.
How did you get interested in your topic? This was a topic that Dr. Hartshorn was much more involved with initially than I was. The project branched off of a Topology / Origami independent study that I had completed with him, and was originally supposed to be much more general—folding and unfolding polyhedra—but it slowly became more focused and specific.
Do you intend to research your topic further? If so, how? I won’t be continuing this line of research, but Dr. Hartshorn has told me that he plans on using it as a jumping-off point for further research with a professor from a local university.
How did you benefit academically by conducting research/participating in honors? I believe the Honors program has prepared me for the kind of work I’ll be doing in the future—large projects, with no expected result, per se. I think it’s a good introduction to the research I’ll most likely be undertaking for the rest of my academic career.
How has the department (or faculty advisor) prepared you for the future? Mathematics is a field that never goes out of style—there is always a demand for mathematicians in business, economics, teaching, research, computing, etc. I think that my mathematical experience and research have prepared me for a wide variety of possible future careers.
What advice do you have for other students interested in Honors? Just do it! If you want to complete a big research project, and there’s a topic you’re interested in—that you could see yourself spending ~10 hours per week on for an entire academic year—there’s no reason you shouldn’t give it a try. If it turns out that the work load is too much, you can always change the project from an Honors project to an independent study, and you still get the research experience, and possibly a publication or two, out of your hard work.
My future plans: I’m waiting to hear back from graduate schools. I’ve applied to Ph.D. programs in astrophysics, and hopefully will be working toward that degree for the next five to seven years.