Intersection the twin dragon with rational lines

Abstract

The Knuth Twin Dragon is a compact subset of the plane with fractal boundary of Hausdorff dimension $s = (\log λ)/(\log \sqrt{2})$, $λ^3 = λ^2 + 2$. Although the intersection with a generic line has Hausdorff dimension $s-1$, we prove that this does not occur for lines with rational parameters. We further describe the intersection of the Twin Dragon with the two diagonals as well as with various axis parallel lines.

Figures (4)

• Figure 1: An automaton characterizing $\partial \mathcal{K}$ (in base $\alpha$).
• Figure 2: The Knuth Twin Dragon $\mathcal{K}$ and its intersection with $\Delta_{1,0,r}$ for some $r$ as in Theorem \ref{['vert']} (red) and with $\Delta_{1,0,-1/5}$ (blue).
• Figure 3: The intersection of $\mathcal{K} = \alpha^{-1} \left(\mathcal{K} \cup (\mathcal{K}+1)\right)$ with lines $\Delta_{0,1,r/2}$, $\Delta_{1,1,-r}$, and $\Delta_{1,-1,r/2}$ for some $r$ as in Theorem \ref{['vert']}.
• Figure 4: Automaton recognizing the imaginary parts of points in $\partial \mathcal{K} \cap \Delta_{1,0,-1/5}$ in base $-4$.

Theorems & Definitions (17)

• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Remark 2.6
• Theorem 3.1