On the 3-dimensional k-Fibonacci spirals

Abstract

The 3-dimensional k-Fibonacci spirals are studied from a geometric point of view. These curves appear naturally from studying the k-Fibonacci numbers {F-k,F-n}(n=0)(infinity) and the related hyperbolic k-Fibonacci functions. In this paper, after a summary of the main properties for the k-Fibonacci numbers, we focus on the geometry features (curvature and torsion) of the 3-dimensional k-Fibonacci spirals. Finally, the Metallic Shofars and their projections on the coordinate planes are also given.