Fibonacci--Theodorus Spiral and its properties

TL;DR

This paper introduces the Fibonacci--Theodorus spiral, explores its geometric properties related to Fibonacci numbers and the golden ratio, and proves a conjecture connecting area ratios to Fibonacci-based lengths.

Contribution

We define a new Fibonacci-based spiral, analyze its properties, and provide a formal proof for Hahn's conjecture on area ratios, extending classical geometric concepts.

Findings

The ratio of two consecutive areas relates to the golden ratio.

Sum of the first n areas approaches a multiple of the sum of the first n Fibonacci numbers.

Proof of Hahn's conjecture on area ratios in the Fibonacci--Theodorus spiral.

Abstract

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 \emph{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 $\sqrt{F_{n+1}}$ and $\sqrt{F_{n+2}-1}$ and the areas between spines of lengths $\sqrt{F_{n}}$ and $\sqrt{F_{n+1}-1}$ 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.