Square Trisection

TL;DR

This paper reviews the history of the square trisection problem, presents a new solution with ruler-and-compass constructions, and conjectures its optimality, contributing to geometric dissection studies.

Contribution

It introduces a novel solution to the square trisection problem along with three ruler-and-compass constructions, and discusses its potential optimality.

Findings

New solution to the square trisection problem

Three ruler-and-compass constructions provided

Conjecture on the optimality of the solution

Abstract

A square trisection is a problem of assembling three identical squares from a larger square, using a minimal number of pieces. This paper presents an historical overview of the square trisection problem starting with its origins in the third century. We detail the reasoning behind some of the main known solutions. Finally, we give a new solution and three ruler-and-compass constructions. We conclude with a conjecture of optimality of the proposed solution.