The coordinate functions of the Heighway dragon curve

TL;DR

This paper investigates the coordinate functions of the Heighway dragon curve for certain angles and establishes a formula for the box-counting dimension of its graph, revealing fractal geometric properties.

Contribution

It provides a new explicit formula for the box-counting dimension of the dragon curve's graph based on the angle parameter.

Findings

The box-counting dimension depends on the angle  and is given by 1 - (log cos)/log 2.

The dimension varies smoothly with the angle within the specified range.

The work deepens understanding of the fractal geometry of the dragon curve.

Abstract

In this work, we study properties of the coordinate functions $x_\theta$ and $y_\theta$ of the dragon curve associated with the angle $\frac{\pi}{3} < \theta < \frac{5\pi}{3}$, and we prove that the box-counting dimension of its graph is equal to $1 - \frac{\log \cos\alpha}{\log 2}$, where $\alpha = \frac{\pi - \theta}{2}$.