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- W2160932976 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. It is well known that the ratio of successive terms of the Fibonacci sequence approaches the mean, τ = (1+ 5 )/2, in the limit and that the diagonal of a regular pentagon with unit edge has length τ. We show that the Fibonacci sequence can be generalized to characterizing all of the diagonals of regular n-gons for n an odd integer. Furthermore, a geometric sequence in τ is also a Fibonacci sequence and shares all of the algebraic properties inherent in the integer Fibonacci sequence. Similar sequences involving the diagonals of higher order n-gons also have algebraic properties. In fact we shall show that the diagonals form the bias vectors of a field. We shall call these, as STEINBACH (1997) did, golden fields. Products and quotients of the diagonals of an n-gon can be expressed as a linear combination of the diagonals. The results depend strongly on a set of polynomials related to the Fibonacci numbers, and the Lucas polynomials, both of which are related to the Chebyshev polynomials. All of the roots of the Fibonacci polynomials are of the form x = 2cos(kπ/n) while the Lucas polynomials map 2cosA a 2cosmA. As a result, we show that a family of matrices with 0, 1, -1 elements form periodic trajectories when operated on by matrix forms of the Lucas polynomials. We refer to these as Mandelbrot Matrix Operators since the Lucas polynomial L 2 (x) corresponds to the Mandelbrot operator at the extreme left hand point on the real axis," @default.
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- W2160932976 date "2005-03-01" @default.
- W2160932976 modified "2023-09-24" @default.
- W2160932976 title "Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices" @default.
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