Construction of helices from Lucas and Fibonacci sequences

Mário M. Graça

Published 2017-07-30Version 1

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.