Algebraic and analytic structure of Morikawa's sangaku problem

Let $μ(r)$ 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 $μ(r)$ was posed in early nineteenth-century Japan by Morikawa. It was proved by Holly and Krumm (2021) that no expression in radicals exists for $μ(r)$. In this article we show that $μ$ 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 $μ(r)$ centered at $r=1$.

💡 Research Summary

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The paper revisits a classic Japanese sangaku problem posed by Morikawa in the early nineteenth century. The geometric configuration consists of two circles of radii 1 and r ( r ≥ 1) that are tangent to each other and to a common horizontal line L. Inside the curvilinear triangular region bounded by the two circles and the line, one seeks the smallest possible side length μ(r) of a square that touches all three boundaries. Earlier work by Holly and Krumm (2021) proved that μ(r) cannot be expressed by radicals, i.e., there is no closed‑form formula built from rational operations and root extractions. The present article goes beyond this negative result by showing that μ(r) is nevertheless an algebraic function and, consequently, real‑analytic on the interval