Chords of an Ellipse, Lucas Polynomials, and Cubic Equations

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 to write down the roots of the Lucas polynomials.

• MSC: Primary 11B39
• Secondary 51M04
• 01A40

Acknowledgments

Several individuals were involved in calling the authors’ attention to Price’s work: Francis Su, who included the special case of Table 1 in his Harvey Mudd Math Fun Facts [32]; Bowen Kerins and Darryl Yong, who then included the problem of computing this product in a problem set during a summer course at the Park City Mathematics Institute; and Sam Shah, who posted the problem on his blog [25], whereby the authors became acquainted with it. The authors wish to thank Kerins in particular, who was a generous correspondent, and also Thomas Price himself, Tom Edgar, and Harold Edwards, for useful comments and for alerting us to [9, 23], and [14], respectively. We would also like to thank the anonymous referees, whose thoughtful feedback greatly improved the article.

Additional information

Notes on contributors

Ben Blum-Smith

BEN BLUM-SMITH received his Ph.D. in 2017 from the Courant Institute of Mathematical Sciences at New York University, after a decade as a teacher and teacher educator. He is currently a Visiting Academic at the NYU Center for Data Science, and has also taught courses at Eugene Lang College, the Bard Prison Initiative, and the Bard Master of Arts in Teaching program. His research interests lie in invariant theory, algebraic combinatorics, their applications to data science, and connections between mathematics and democracy.

Japheth Wood

JAPHETH WOOD received his Ph.D. in 1997 from the University of California, Berkeley. He has had academic appointments at the Pontificia Universidad Católica de Chile, Vanderbilt University, the University of Louisville, and Chatham University. Since 2006 he has been a faculty member at Bard College in Annandale-on-Hudson, New York, including stints with the Master of Arts in Teaching program and the Bard Prison Initiative. Japheth led the New York Math Circle from 2012 to 2015 and has taught at the Hampshire College Summer Studies in Mathematics program. He is co-director of the Bard Math Circle.