Are 3D Fibonacci spirals for real?: From science to arts and back to science

This paper is aiming to start an investigation on possible extensions of the classical Fibonacci spiral (2D) to three dimensions (3D). Our goal is to understand if and how 3D spirals, which are called or linked to Fibonacci, are in fact supported mathematically. Our inspiration here has been the 2D Padovan spiral for which a 3D extension is known. The classical 2D Fibonacci spiral is generated on a particular tiling of the 2D space with squares which follow a well-known Fibonacci identity. In this paper we shall analyze quite a few 3D spirals. The main conclusion is that the search for a 3D Fibonacci spiral is left wanting, as none of the 3D spirals we were able to find is backed by a Fibonacci identity describing a recurrent tiling of the 3D space (similar to the recurrent tiling in 2D). Still, this paper has set the stage for the construction of 3D Fibonacci spirals by showing that for distinguishing different tilings of the 3D space a thorough understanding of homogenous Fibonacci cubic identities is compulsory.