Interesting Esoterica

Chords of an ellipse, Lucas polynomials, and cubic equations

Article by Ben Blum-Smith and Japheth Wood
  • Published in 2018
  • Added on
A beautiful result of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse. We give a conceptually transparent development of this result that provides a tour of several gems of classical mathematics: It is inspired by Girolamo Cardano's solution of the cubic equation, uses Newton's theorem connecting power sums and elementary symmetric polynomials, and yields for free an alternative proof of the Binet formula for the generalized Lucas polynomials.

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BibTeX entry

@article{ChordsofanellipseLucaspolynomialsandcubicequations,
	title = {Chords of an ellipse, Lucas polynomials, and cubic equations},
	abstract = {A beautiful result of Thomas Price links the Fibonacci numbers and the Lucas
polynomials to the plane geometry of an ellipse. We give a conceptually
transparent development of this result that provides a tour of several gems of
classical mathematics: It is inspired by Girolamo Cardano's solution of the
cubic equation, uses Newton's theorem connecting power sums and elementary
symmetric polynomials, and yields for free an alternative proof of the Binet
formula for the generalized Lucas polynomials.},
	url = {http://arxiv.org/abs/1810.00492v3 http://arxiv.org/pdf/1810.00492v3},
	year = 2018,
	author = {Ben Blum-Smith and Japheth Wood},
	comment = {},
	urldate = {2019-06-30},
	archivePrefix = {arXiv},
	eprint = {1810.00492},
	primaryClass = {math.HO},
	collections = {easily-explained,fun-maths-facts,puzzles}
}