The Heighway dragon revisited (Q1404525)

Suppose \(P_0=\{(0,y):0\leq y\leq 1\} \subset\mathbb{R}^2\), \(P_1\) is obtained by replacing \(P_0\) with two line segments of equal length \(1/ \sqrt 2\) jointly at a right angle, with \(P_1\) on the left of \(P_0\), and with the same two ends as \(P_0\). \(P_n\) is obtained by iterating \(P_{n-1}\) left and right alternately, starting with left, counting from the lower endpoint \((0,0)\). \(\{P_n\}\) is a Cauchy sequence of compact sets in \(\mathbb{R}^2\) in the Hausdorff metric. The limit of \(\{P_n\}\) as \(n\to\infty\) is called the Heighway dragon. The authors proved that the Heighway dragon is the closure of a countable union of geometrically similar closed disk-like planar sets which intersect each other in a linear order: any two of them intersect at no more than one point and for any three disks there exist at least two with an empty intersection.