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My honors thesis looks at various applications of origami in mathematics. I’m doing this through a number of different ways, each of which will comprise a different chapter.
The first chapter that I’m writing involves proving a particular theorem published by two mathematicians. It states that given a group of straight lines on a piece of paper – whether they be arranged into shapes, pictures, words, just scattered, or any other configuration – can be folded up so that all the lines, and only the lines, can be cut out with a single, straight cut. Moreover, the theorem goes on to give two algorithms with which this process can be practically done. Right now, I’m working on exploring the theorem, proving it, and finding all sorts of special cases that need explanation.
Additionally, I’m working on researching a few other topics to see what I can discover about various mathematical concepts through origami. For instance, a fairly well-known way of creating a hyperbolic paraboloid (a doubly ruled surface shaped like a saddle) is by folding alternating mountain and valley folded concentric squares on a square piece of paper. I began to wonder if folding different shaped paper (say, octagons or dodecagons) would still produce a hyperbolic parabolic or something entirely different. In fact, they produce shapes that look a bit like a hyperbolic paraboloid, but have some fundamental differences. I’m now working on trying to develop precise equations to define my shapes.
Another topic I’ve been discussing with my adviser is creating various fractions using origami paper. For instance, using a square piece of paper with each side as one unit, it is fairly easy to get a measurement of one-half (fold the paper in half). To get an accurate measurement for one-third is a bit trickier, but still quite doable. A number of different fractions have actually been solved for, but I’m looking into possibly developing an algorithm for folding all possible fractions.
Working on an individual basis with my adviser, Professor Berkove, has been great because we get to cover all sorts of topics in math in a variety of different areas according to what interests me most and what I find most relevant to my work. He has introduced me to a number of new areas in math, and has helped me immensely in discovering avenues for answering the questions that both he and I have posed regarding my work.
After graduation, I’m hoping to get a job that incorporates both my mathematics and English backgrounds. I’m currently looking into a number of different fields that might offer me that opportunity. The main benefit that my thesis has given me toward that goal is the courage to pursue my own creative interests and practice in utilizing the various individual tools I have been acquiring through my four years here into one main project. That goal has only been encouraged by Professor Berkove, who pushes me to find ways of answering the questions I pose with my own abilities.