Fibonacci--Theodorus Spiral and its properties

Inspired by the ancient spiral constructed by the greek philosopher Theodorus which is based on concatenated right triangles, we have created a spiral. In this spiral, called Fibonacci--Theodorus, the sides of the triangles have lengths corresponding to Fibonacci numbers. Towards the end of the paper, we present a generalized method applicable to second-order recurrence relations. Our exploration of the Fibonacci--Theodorus spiral aims to address a variety of questions, showcasing its unique properties and behaviors. For example, we study topics such as area, perimeter, and angles. Notably, we establish a relationship between the ratio of two consecutive areas and the golden ratio, a pattern that extends to angles sharing a common vertex. Furthermore, we present some asymptotic results. For instance, we demonstrate that the sum of the first n areas comprising the spiral approaches a multiple of the sum of the initial n Fibonacci numbers. Moreover, we provide a sequence of open problems related to all spiral worked in this paper. Finally, in his work Hahn, Hahn observed a potential connection between the golden ratio and the ratio of areas between spines of lengths √Fₙ₊₁ and √Fₙ₊₂-1 and the areas between spines of lengths √Fₙ and √Fₙ₊₁-1 in the Theodorus spiral. However, no formal proof has been provided in his work. In this paper, we provide a proof for Hahn's conjecture.