Construction of helices from Lucas and Fibonacci sequences

TL;DR

This paper introduces a novel method to construct and analyze helices derived from Lucas, Fibonacci, and Pell sequences using complex functions, revealing invariant ratios and new geometric properties.

Contribution

It presents a new approach to generate helices from Lucas, Fibonacci, and Pell sequences and explores their geometric and algebraic properties, including invariant ratios and Pell's equations.

Findings

Helices constructed from these sequences have specific ratio properties.

Pell's helix ratio is invariant, unlike P-Lucas helices.

Linear combinations of complex maps generate new related helices.

Abstract

By means of two complex-valued functions (depending on an integer parameter P>=1) we construct helices of integer ratio R>=1 related to the so-called Binet formulae for P-Lucas and P-Fibonacci sequences. Based on these functions a new map is defined and we show that its three-dimensional representation is also a helix. After proving that the lattice points of these later helix satisfy certain diophantine Pell's equations we call it a Pell's helix. We prove that for P-Fibonacci and Pell's helices the respective ratio is an invariant, contrasting to the P-Lucas helices whose ratio depends on P. It is also shown that suitable linear combinations of certain complex-valued maps lead to new helices related to Lucas/Fibonacci/Pell numbers. Graphical examples are given in order to illustrate the underlying theory.