Can the quadratrix truly square the circle?

TL;DR

This paper investigates whether the quadratrix can square the circle without limits by translating the problem into algebra and number theory, ultimately concluding it cannot due to transcendental number theory constraints.

Contribution

It reformulates the circle squaring problem via algebraic analogies and connects it to open questions in transcendental number theory, providing new insights into the quadratrix's limitations.

Findings

The quadratrix cannot square the circle without limits.

The problem relates to the transcendence of π and algebraic properties.

It links geometric constructions to open conjectures in number theory.

Abstract

The quadratrix received its name from the circle quadrature, squaring the circle, but it only solves it if completed by taking a limit, as pointed out already in antiquity. We ask if it can square the circle without limits and restrict its use accordingly, to converting ratios of angles and segments into each other. The problem is then translated into algebra by analogy to straightedge and compass constructions, and leads to an open question in transcendental number theory. In particular, Lindemann's impossibility result no longer suffices, and the answer depends on whether $\pi$ belongs to the analog of Ritt's exponential-logarithmic field with an algebraic base. We then derive that it does not from the well-known Schanuel conjecture. Thus, the quadratrix so restricted cannot square the circle after all.