Folding the Heighway dragon curve

TL;DR

This paper explores the equivalence of two construction methods for the Heighway dragon fractal curve, generalizing the process and proving their equivalence through algebraic properties.

Contribution

It introduces a generalized construction allowing rotations on both sides and proves the equivalence of the methods using a duality theorem and operator distributivity.

Findings

The two construction approaches are equivalent under the generalized framework.

The key property for proof is the distributivity of the interleave operator.

Generalization includes rotations to both sides of the 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.