2020/04/19 by Bernhard R. Parodi, Parodi, Bernhard R.
Materials Science · Mathematics · Physics and Astronomy · #11B39 #14H50 #2010: 11B37 #39A06 #65Q30 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #History and Overview (math.HO) #Mathematics and Applications #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2004.08902
openalex publication_date 2020/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\inR, exhibiting spiral graphs and oscillating curves in the Gaussian plane, subsumes the values Gn for t\inN0 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.