The Illustrating Mathematics Program at ICERM
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The Illustrating Mathematics Program at ICERM RICHARD EVAN SCHWARTZ
This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of ‘‘mathematical community’’ is the broadest: ‘‘schools’’ of mathematics, circles of correspondence, mathematical societies, student organizations, extra-curricular educational activities (math camps, math museums, math clubs), and more. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
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his article is an account of the semester-long program held at Brown University’s Institute for Computational and Experimental Research in Mathematics (ICERM) in fall 2019. I was one of the organizers of the program, and I also taught a class there, called Geometry and Illustration. I don’t want to mislead you into thinking that somehow I am at the center of what is going on in the program, but let me explain first of all what it is like to teach a class full of intensely creative professors, postdocs, and graduate students who love illustrating mathematics. My main goal for the course was to present topics in geometry, topology, dynamics, and discrete groups that are especially amenable to visual thinking. Much has been said already about the powerful role of visual thinking in the process of discovery in geometry and topology, and I wanted to put that into practice. When I planned the course, I hoped that the class participants would bring to bear the visual tools they were developing in the program in exploring the topics of the class, discover new things about them, and understand them in new ways. The first day of the course, I talked about the spin cover, which at bottom is a beautiful way to represent rigid motions of the sphere using quaternions. During the lecture, I demonstrated the famous ‘‘belt trick,’’ in which you sort of undulate your arm so that your palm spins around twice. Some weeks later, Alexander Holroyd, one of the participants in the program, showed me a robotic arm he had made out of Legos that did the belt trick in an ingenious way. Holroyd’s construction (Figure 1) is based on a chain of universal joints arranged in what looks like an octagonal question mark. It is a geometry theorem written in plastic and metal. A few lectures later, I told the class about the Penrose tilings, planar tilings by kite-shaped and dart-shaped pieces that have a hierarchical structure: a tiling at one scale determines, by a process called inflation, a tiling at a larger scale that you can superimpose over the initial one. This larger-scale tiling in turn determines a tiling at a still larger scale, and so on. The reason that a Penrose tiling cannot have an infinitely repeating pattern in it is that such a pattern would of necessity have a scale to it—-say a repetition
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