Cyclotomic Points and Algebraic Properties of Polygon Diagonals

TL;DR

This paper explores the algebraic and number-theoretic properties of diagonals in regular polygons by linking geometric ratios to roots of unity and polynomial evaluations, providing new classifications and generalizations.

Contribution

It introduces algorithms for analyzing polygon diagonals using roots of unity and classifies when metallic ratios can be expressed as diagonal ratios, extending to number field degree analysis.

Findings

Classified when metallic ratios can be realized as diagonal ratios

Developed algorithms for polynomial evaluations at roots of unity

Generalized to the degree of number fields generated by diagonal ratios

Abstract

By viewing the regular $N$-gon as the set of $N$th roots of unity in the complex plane we transform several questions regarding polygon diagonals into when a polynomial vanishes when evaluated at roots of unity. To study these solutions we implement algorithms in Sage as well as examine a trigonometric diophantine equation. In doing so we classify when a metallic ratio can be realized as a ratio of polygon diagonals, answering a question raised in a PBS Infinite Series broadcast. We then generalize this idea by examining the degree of the number field generated by a given ratio of polygon diagonals.