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Folding the Heighway dragon curve

Ting-Wu Chang, Liang-Ting Chen, Shin-Cheng Mu

Abstract

The Heighway dragon curve is one of the most known fractal curves. There are two ways to construct the curve: repeatedly make a copy of the current curve, rotate it by 90 degrees, and connect them; or repeatedly replace each straight segment in the curve by two segments with a right angle. A natural question is how do we prove the equivalence of the two approaches? We generalise the construction of the curve to allow rotations to both sides. It then turns out that the two approaches are respectively a foldr and a foldl, and the key property for proving their equivalence, using the second duality theorem, is the distributivity of an "interleave" operator.

Folding the Heighway dragon curve

Abstract

The Heighway dragon curve is one of the most known fractal curves. There are two ways to construct the curve: repeatedly make a copy of the current curve, rotate it by 90 degrees, and connect them; or repeatedly replace each straight segment in the curve by two segments with a right angle. A natural question is how do we prove the equivalence of the two approaches? We generalise the construction of the curve to allow rotations to both sides. It then turns out that the two approaches are respectively a foldr and a foldl, and the key property for proving their equivalence, using the second duality theorem, is the distributivity of an "interleave" operator.

Paper Structure

This paper contains 5 sections, 1 theorem, 8 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Formalisation
  3. The proof
  4. Generalization
  5. Conclusions and related work

Key Result

Theorem 1

$\mathit{foldr}\;(\oplus)\;\mathit{e}\;\mathit{xs}\mathrel{=}\mathit{foldl}\;(\otimes)\;\mathit{e}\;\mathit{xs}$ for all finite list $\mathit{xs}$ if

Figures (5)

  • Figure 1: Dragon curves of order 8 and 12.
  • Figure 2: The unfolding construction of dragon curves of orders 0 to 4. The rotated copy is shown in thick lines, and the old endpoint (the center of rotation) is shown as an empty arrow.
  • Figure 3: The folding construction of dragon curves of orders 0 to 4. The previous curves for each order is shown using dotted lines.
  • Figure 4: Dragon curve creation and paper folding.
  • Figure 5: Generalised dragon curves.

Theorems & Definitions (1)

  • Theorem 1