Making Math Material

ost mathematical topics, ideas, and formulas that we encounter in a math class live in an abstract, symbolic world. We write equations, draw diagrams, and even solve practical-sounding problems, all confined to marks on paper or perhaps to electronic representations. But what happens when we try to embody some of those concepts in physical form? In fall 2018, we, a group of students at Harvard University with diverse academic backgrounds and our instructor, Glen Whitney, had the opportunity to explore this question in a course entitled, “Making Math Material.” The course proceeded in a cyclic fashion: we took up a mathematical topic of interest and studied it in some depth to discover the entities and ideas that could be illustrated physically. Then we adopted a software tool—GeoGebra, SymmetriSketch, OpenSCAD, and others—suited to creating virtual models of the objects we might want to build. Once we identified interesting targets, we explored fabrication methods to turn these designs into reality, ending with tangible models we could touch, manipulate, and explore. In doing so, new or additional mathematical questions arose, kicking off another round of study, design, and building. Creating models served other purposes as well: to make the mathematics we were studying accessible to a larger audience and to make it more relevant to them. The major projects of the semester highlighted these goals: writing a blog about the mathematics found in everyday objects, designing an item that would bring out the beauty of a mathematical topic, and finally, creating a handson exhibit suitable for an interactive museum. The following pages contain a selection of these projects described by the students who created them. Like any offbeat pursuit, the course also had other outcomes. For example, we became more adept with utility knives. But among the most satisfying and exciting outcomes were the unforeseen discoveries in the course of physical exploration. Lia Mondavi uncovered an aspect of a geometric proof of a number-theoretic identity that seemed to be overlooked in other presentations; Katja Diaz-Granados found that a principle elucidated by Leonardo da Vinci held up surprisingly well under modern measurements; and Jacob Bindman discovered a three-dimensional linkage that would contract along one axis when either compressed or stretched along a different axis. Hopefully these experiences and discoveries will inspire others to explore the material side of mathematics.