The Heighway Dragon Revisited

We prove that the Heighway dragon is a countable union of closed geometrically similar disk-like planar sets which intersect each other in a linear order: any two of them intersect at no more than one cut point and for any three disks there exist at least two with an empty intersection. Consequently, the interior of the Heighway dragon is a countable union of disjoint open disk-like planar sets. We determine all the cut points of the dragon and show that each disk-like subset between two cut points is a graph self-similar set denned by a graph-directed iterated function system consisting of four seed sets. Our results describe a fairly complete picture of the topological and geometric structure of the Heighway dragon.