A novel approach to the Fibonacci numbers by inverse stereographic projection

Abstract

In this study, a new geometric framework for Fibonacci numbers is introduced through the definition of Fibonacci points, whose components are Fibonacci numbers. These points are then mapped onto the sphere by means of inverse stereographic projection (ISP), providing novel recurrence relations and algebraic structures analogous to those in classical ones. The transformation of Fibonacci-related geometrical entities such as lines and circles is investigated, leading to the definition of new spherical regions associated with Fibonacci parallels and meridians. Furthermore, a deformed version of the Binet–Fibonacci spiral passing through the Fibonacci points is constructed, which may offer applications in modeling biological spiral deviations caused by environmental effects. Classical Fibonacci identities, such as Cassini identities, are revisited in this spherical setting, and their transformations are established. Finally, possible generalizations of these results to higher-dimensional spaces are discussed.