Optimization and Optimal Control Approach to Japanese Sangaku Six Equal Circles Problem

Inspired by a Japanese sangaku problem, we examine six equal circles problems from a view point of sphere packing theory. The original problem was to find the size of the rectangular tablet within which nonoverlapping six circles of the same radius are arranged in a particularly nice configuration. Knowing its solution for the rectangle size, we formulate an optimization problem to ask conversely if we can reconfigure the six circles to maximize the area that they cover within the rectangle. This optimization problem is to maximize a quadratic convex function subject to nonconvex constraints. An attempt to find locally optimal solutions for the optimization problems has numerically demonstrated that the original sangaku configuration is indeed a local optimum and that there is another local optimum which covers a larger area. We formulate a new optimal control problem based on sangaku problem.