Folding the dragon

Abstract

The Heighway Dragon Curve is one of the best known fractal curves. There are two ways to construct the curve: starting from a straight line segment, 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 that the key property for proving their equivalence, using the second duality theorem, is the distributivity of an `interleave' operator.