Hausdorff Dimension of Generalized Fibonacci Word Fractals

TL;DR

This paper computes the Hausdorff dimension of the scaling limits of Fibonacci word fractals, revealing their fractal complexity for various Fibonacci curves and drawing angles.

Contribution

It introduces a method to determine the Hausdorff dimension of generalized Fibonacci word fractals for different angles and Fibonacci variants.

Findings

Hausdorff dimension varies with Fibonacci curve type

Dimension computed for angles between 0 and π/2

Provides insights into fractal complexity of Fibonacci curves

Abstract

Fibonacci word fractals are a class of fractals that have been studied recently, though the word they are generated from is more widely studied in combinatorics. The Fibonacci word can be used to draw a curve which possesses self-similarities determined by the recursive structure of the word. The Hausdorff dimension of the scaling limit of the finite Fibonacci word curves is computed for $i$-Fibonacci curves and any drawing angle between $0$ and $\pi/2$.