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Algebraic and analytic structure of Morikawa's sangaku problem

David Krumm

Abstract

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 $r=1$.

Algebraic and analytic structure of Morikawa's sangaku problem

Abstract

Let denote the minimal side length of a square inscribed in the curvilinear triangular region formed by two tangent circles of radii and together with their common tangent line. The problem of finding a closed-form expression for was posed in early nineteenth-century Japan by Morikawa. It was proved by Holly and Krumm (2021) that no expression in radicals exists for . In this article we show that is an algebraic function, and consequently real-analytic on 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 , whose coefficients may be computed by Newton iteration from the defining algebraic equation. We illustrate the method by explicitly computing the Taylor expansion of centered at .
Paper Structure (4 sections, 6 theorems, 23 equations, 1 figure, 1 table)

This paper contains 4 sections, 6 theorems, 23 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. An algebraic equation for the Morikawa function
  3. Analyticity outside a finite set
  4. Computation of a Taylor expansion

Key Result

Theorem 1.1

There exists a finite set $E\subset[1,\infty)$ of real algebraic numbers such that the function $\mu\colon [1,\infty)\setminus E \longrightarrow\mathbb{R}$ is real-analytic.

Figures (1)

  • Figure 1: The region determined by two tangent circles of radii $1$ and $r\ge1$ together with their common tangent line, and an inscribed square of minimal side length $\mu(r)$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2: Real-analytic implicit function theorem
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • proof