Non-self-intersective dragon curves

TL;DR

This work establishes a rigorous, threshold-based non-self-intersection result for dragon curves generated by uniform folding. By modeling the limit Dragon curve $\mathcal{D}(\xi)$ as the attractor of an IFS with $f_1$ and $f_2$ and employing a renormalization approach, the authors construct an infinite polygon $\mathcal{C}$ and verify an open-set-type framework, including disjointness of the IFS images, to prove $\mathcal{D}(\xi)$ is a simple arc for $0<\xi<\xi_0$, where $\xi_0\approx 0.703858$. This corresponds to unfolding angles $\theta=\pi-2\xi$ exceeding approximately $99.3438^{\circ}$, and they extend the result to renormalized dragon curves $D_k$, showing no self-intersections for the same $\xi$ range. The paper also outlines a finite-polygon truncation method to prove non-self-intersection for $D_k$ and discusses open problems, including the precise supremum of $\xi$ and the behavior beyond $\xi_0$ and $\pi/3$.

Abstract

Let us fold a strip of paper many times in the same direction, and then unfold it to form a fixed angle $θ$ at all creases. The resulting shape is called the Dragon curve with the unfolding angle $θ$. When $0\leθ<90^{\circ}$, the corresponding Dragon curve has a self-intersection. When $θ=180^{\circ}$, the corresponding Dragon curve is a straight line, which has no self-intersection. In this paper, we will show that any Dragon curve whose unfolding angle is greater than $99.3438^{\circ}$ and less than $180^{\circ}$ has no self-intersection.

Figures (17)

• Figure 1: Folding-unfolding process for $k=1,2$
• Figure 2: The Heighway Dragon of order 8
• Figure 3: The limit Dragon curve ${\mathcal{D}}(0.703858)$ is a simple arc.
• Figure 4: The sets $A_{1}$ and $\widetilde{A_{1}}$
• Figure 5: The infinite polygon $\mathcal{C}$

Theorems & Definitions (33)

• Definition 1
• Proposition 1
• Theorem 1
• Theorem 2
• Definition 2
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Definition 3