Sangaku: A Mathematical, Artistic, Religious, and Diagrammatic Examination

Abstract

Between the 17th and 19th centuries, mathematically orientated votive tablets appeared in Shinto shrines and Buddhist temples all over Japan. Known as sangaku, they contained problems of a largely geometrical nature. The 18th century also saw the Japanese mathematician Seki Takakazu develop a form of algebra known as tenzan jutsu. To date, much of the literature regarding sangaku employs modern mathematical notation and techniques to solve their problems. In this research I solve problems from nine sangaku using modern techniques. As well as this, I also illustrate how sangaku can be solved using the traditional Japanese tenzan jutsu of Seki. I compare seven problems taken from sangaku with similar problems solved using tenzan jutsu from the 1810 Japanese text the Sanpo Tenzan Shinan. I show the tenzan jutsu technique can be applied to solve the sangaku problems. The question of whether these tablets had purely communicative and mathematical functions is also still debated by researchers. In this thesis I argue that an examination of the creation, location, and writings on these tablets by their authors shows sangaku had multiple functions and should be considered artistic, religious, as well as mathematical artefacts. I also examine the role of diagrams on sangaku, and argue that they are a vital element of determination for the problems on these tablets.