On the Construction of 3D Fibonacci Spirals

Mariana Nagy (), Simon R. Cowell and Valeriu Beiu
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Mariana Nagy: Department of Mathematics and Computer Science, “Aurel Vlaicu” University of Arad, 2-4 Elena Drăgoi Str., RO-310330 Arad, Romania
Simon R. Cowell: Department of Mathematics and Computer Science, “Aurel Vlaicu” University of Arad, 2-4 Elena Drăgoi Str., RO-310330 Arad, Romania
Valeriu Beiu: Department of Mathematics and Computer Science, “Aurel Vlaicu” University of Arad, 2-4 Elena Drăgoi Str., RO-310330 Arad, Romania

Mathematics, 2024, vol. 12, issue 2, 1-19

Abstract: The paper aims to extend the classical two-dimensional (2D) Fibonacci spiral into three-dimensional (3D) space by using geometric constructions starting from cubic Fibonacci identities and relying on affine maps and parametrizations of the curves. We have already performed a comprehensive survey of cubic Fibonacci identities, which, to our surprise, uncovered only a handful of homogenous cubic identities. Obviously, the goal here is to show how one could use a particular homogenous cubic Fibonacci identity for generating 3D geometric designs similar in spirit to the way the classical Fibonacci spiral is built in 2D starting from a quadratic Fibonacci identity. This made us realize that for any cubic identity there are many different ways of packing cuboids, while only an insignificant fraction of those possible tilings might allow a smooth spiral-like curve to be drawn through them. After reviewing the state of the art, we present accurate details on ways to construct such 3D spirals using affine maps. We go on to prove the continuity and smoothness of such 3D spirals by giving a parametrization of the intersection of the surfaces that define the curves. Throughout the paper, we visualize the resulting 3D spirals by generating geometrically correct stereoscopic views. Finally, it is to be mentioned that the recursive 3D packing of cuboids tends to lead to fractal structures, which will need further investigations.

Keywords: Fibonacci sequence; Fibonacci spiral; Padovan sequence; fractals; affine maps; projective geometry (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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