Shifted L\'evy's Dragon Curve and Directed Graph

TL;DR

This paper introduces directed graphs that characterize the complex power series representations of Levy's Dragon Curve and its translation, revealing structural similarities.

Contribution

It constructs and analyzes two directed graphs that encode the power series representations of the original and translated Levy's Dragon Curve.

Findings

Graph $4a1$ characterizes the original curve's representation.

Graph $4a2$ characterizes the translated curve.

Both graphs exhibit revolving structures with analogous properties.

Abstract

It is known that every point on L\'evy's Dragon Curve admits a natural representation as a complex power series. We introduce a directed graph $\mathcal{G}_1$ which characterizes this representation. In this paper, we study the translation of the curve by $s=-1/2+i/2$. We identify another directed graph $\mathcal{G}_2$, that characterizes the translated curve and exhibits a revolving structure analogous to that of $\mathcal{G}_1$.