Chords of an ellipse, Lucas polynomials, and cubic equations

TL;DR

This paper explores the deep connections between ellipse geometry, Lucas and Fibonacci polynomials, and classical algebraic solutions, providing new insights and interpretations rooted in historical mathematical theorems.

Contribution

It offers a reorganized proof of Price's theorem linking Fibonacci and Lucas polynomials to ellipse geometry, with new ideas connecting to symmetric polynomials and classical algebra.

Findings

Generalized Lucas polynomials relate to symmetric polynomial theory.

Cardano's method can explicitly determine roots of Lucas polynomials.

Connections between Fibonacci polynomials and ellipse geometry are established.

Abstract

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse, generalizing a classic problem about circles. We give a brief history of the circle problem, an account of Price's ellipse proof, and a reorganized proof, with some new ideas, designed to situate the result within a dense web of connections to classical mathematics. It is inspired by Cardano's solution of the cubic equation and Newton's theorem on power sums, and yields an interpretation of generalized Lucas polynomials in terms of the theory of symmetric polynomials. We also develop additional connections that surface along the way; e.g., we give a parallel interpretation of generalized Fibonacci polynomials, and we show that Cardano's method can be used write down the roots of the Lucas polynomials.