A generalized Fibonacci spiral
Abstract
As a generalization of planar Fibonacci spirals that are based on the recurrence relation Fn=Fn-1+Fn-2, we draw assembled spirals stemming from analytic solutions of the recurrence relation Gn=a\, Gn-1+b\, Gn-2+c\, d\,n, with positive real initial values G0 and G1 and coefficients a, b, c, and d. The principal coordinates given in closed-form correspond to finite sums of alternating even- or alternating odd-indexed terms Gn. For rectangular spirals made of straight line segments (a.k.a. spirangles), the even-indexed and the odd-indexed directional corner points asymptotically lie on mutually orthogonal oblique lines. We calculate the points of intersection and show them in the case of inwinding spirals to coincide with the point of convergence. In the case of outwinding spirals, an n-dependent quadruple of points of intersection may form. For arched spirals, interpolation between principal coordinates is performed by means of arcs of quarter-ellipses. A three-dimensional representation is exhibited, too. The continuation of the discrete sequence \Gn\ to the complex-valued function G(t) with real argument t∈R, exhibiting spiral graphs and oscillating curves in the Gaussian plane, subsumes the values Gn for t∈N0 as the zeros. Besides, we provide a matrix representation of Gn in terms of transformed Horadam numbers, retrieve the Shannon product difference identity as applied to Gn, and suggest a substitution method for finding a variety of other identities and summations related to Gn.