Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices

Jay Kappraff¹*, Slavik Jablan², Gary W. Adamson³ and Radmila Sazdanovich²

¹New Jersey Institute of Technology, Newark, NJ 07102, U.S.A.
²The Mathematical Institute, Knez Mihailova 35, P.O. Box 367, 11001, Belgrade, Serbia and Montenegro
³P.O. Box 124571, San Diego, CA 92112-4571, U.S.A.
*E-mail address: kappraff@verizon.net

(Received March 11, 2005; Accepted March 20, 2005)

Keywords: Fibonacci Sequence, Golden Mean, Regular Polygons, Diagonals, Lucas Polynomials, Chebyshev Polynomials, Mandelbrot Set

Abstract. The diagonals of regular n-gons for odd n are shown to form algebraic fields with the diagonals serving as the basis vectors. The diagonals are determined as the ratio of successive terms of generalized Fibonacci sequences. The sequences are determined from a family of triangular matrices with elements either 0 or 1. The eigenvalues of these matrices are ratios of the diagonals of the n-gons, and the matrices are part of a larger family of matrices that form periodic trajectories when operated on by a matrix form of the Mandelbrot operator at a point of full-blown chaos. Generalized Mandelbrot matrix operators related to Lucas polynomials have similar periodic properties.