Hiroshi Kawabi: Are You Working Things Out by Hand?

2018/11/08

My specialty is mathematics, and I mainly research probability theory. I verify ideas that arise during research and perform detailed calculations in notebooks or on whiteboards in my office. Also, when learning new things through other people's papers, it is rarely possible to understand just by following the words on the page. So, for the time being, I move my hands, copying definitions, theorems, and proofs into my notebook to look at them, or doing small calculations of simple examples. For me, it seems that the act of moving my hands and writing on paper is closely connected to the act of thinking.

As one of my research themes in recent years, I have been investigating the long-term behavior of random walks on graphs with periodicity, such as crystal lattices (with triangular and hexagonal lattices as typical examples), together with a geometer friend from my student days. Until this research got on track, we frequently traveled between each other's universities for discussions, and I quickly noticed that the stroke order for drawing a hexagonal lattice on the blackboard was significantly different between him and me. While I would first draw one hexagon and then add copies translated vertically, he would first draw a square lattice, place a point inside each square, and then connect those points to three vertices of the surrounding squares to draw the hexagonal lattice. At first, I thought what a strange stroke order he was using, but as the research progressed, I realized that this very stroke order was the natural way to understand "harmonic realization," which is the "geometrically most beautiful realization" of a crystal lattice, and it felt like the scales had fallen from my eyes. This is a trivial detail that would not have emerged just by following the words in papers or books, but was something I understood after moving my hands in various ways through discussions with others. I was surprised at how much a difference in stroke order could completely change one's perspective.

At Okayama University, where I worked for 11 years until this March, I gave lectures to students in the Department of Mathematics in the Faculty of Science. When introducing complex figures or new mathematical concepts, I paid close attention to the stroke order on the blackboard. Currently, I teach mathematics to students in the Faculty of Economics at Hiyoshi. As expected, the student temperament is quite different, and I get the impression that many students want to study mathematics efficiently as a tool for describing economics. I will continue to use trial and error to aim for lectures where even students in the Faculty of Economics can see the free mathematical thinking behind the words in the textbook by moving their hands.