Sangaku Optimization Problems: An Algebraic Approach
During the Edo Period (1603-1867), Japan was isolated from the influence of western mathematics. Despite this isolation, Japanese mathematics, called Wasan, flourished, and a unique approach to present mathematical problems was developed. Painted wooden tablets called sangaku were hung on display at Shinto shrines and Buddhist temples for recreational enjoyment and religious offerings. More than 900 tablets have been discovered with problems developed by priests, samurai, farmers, and children. The vast majority of these problems were solved using analytic geometry and algebraic means, and the collection as a whole is frequently referred to as Japanese Temple Geometry. Within the collection of the sangaku, several optimization problems appear with answers included. However, the methods used to obtain those answers are absent. Because the work of Newton and Leibniz was unknown to the Japanese mathematicians of that time and no evidence exists of contemporaneous Japanese mathematicians having a formal definition of the derivative, their solution techniques to these problems remains unresolved (Fukagawa and Rothman 2008). To illustrate a possible noncalculus approach for the solution to sangaku optimization problems, we will examine two specific examples. To help readers visualize the two examples, Maple™ animations have been created by the authors and can be found at http://www.mesacc.edu/~davvu41111/Sangaku.htm.