Algebraic and analytic structure of Morikawa's sangaku problem

Abstract

Let $\mu(r)$ denote the minimal side length of a square inscribed in the curvilinear triangular region formed by two tangent circles of radii $1$ and $r \ge 1$ together with their common tangent line. The problem of finding a closed-form expression for $\mu(r)$ was posed in early nineteenth-century Japan by Morikawa. It was proved by Holly and Krumm (2021) that no expression in radicals exists for $\mu(r)$. In this article we show that $\mu$ is an algebraic function, and consequently real-analytic on $[1,\infty)$ outside a finite explicitly computable set. In particular, although no expression in radicals exists, the function admits convergent Taylor expansions at all non-exceptional values of $r$, whose coefficients may be computed by Newton iteration from the defining algebraic equation. We illustrate the method by explicitly computing the Taylor expansion of $\mu(r)$ centered at $r=1$.