Iterated Function
Systems
Larry Riddle

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Lévy Dragon

Paul Levy

Construction
Animation
(17Kb)

Start with an isoceles right triangle L0. Replace this triangle with two isoceles right triangles so that the hypotenuse of each new triangle lies on one of the equal sides of the old triangle. Place each new triangle so that it points out from the original triangle to get L1. For the next step, repeat the process on each of the triangles that make up L1, as illustrated in the figure below.

For each new iteration, replace each triangle in Lk by two isoceles right triangles. The Levy dragon is the limiting set of this iterative construction. Notice that at each iteration, the sides of each triangle in Lk are scaled by a factor 1/sqrt(2) [see IFS discussion below], so that the area is scaled by 1/2. But each triangle in Lk produces two new triangles, so the total area remains unchanged. Thus the area of the Lévy dragon is the same as the area of the original isoceles right triangle L0.

Suppose that the original isoceles right triangle L0 is placed so that its hypotenuse is the unit interval on the x-axis and the opposite corner is at (1/2, 1/2). Then each leg is equal to

. This means that we must scale the triangle L0 by the factor r =

to get each of the new triangles for L1.

One triangle must then be rotated by 45°, while the other triangle must be rotated by -45° (i.e. in the clockwise direction) and translated by 1/2 in both the x and y directions. This yields the following IFS

where r = . The fixed invariant set of this IFS will be the Lévy dragon.

Angle 45
Axiom F
F --> + F - - F +

First four iterations of the L-system

The line segments generated by the L-system correspond to the hypotenuses of the isoceles right triangles generated in the construction described above.

We have a hyperbolic IFS with each map being a similitude of ratio r < 1. Therefore the similarity dimension, d, of the unique invariant set of the IFS is the solution to

Paul Lévy studied the curve now known as Lévy's Dragon as part of a general study of curves consisting of parts similar to the whole [see the original 1938 article by Levy or the translation in Edgar]. In this he was motivated by the earlier work of Helge von Koch and the Koch curve.

Among several of the properties that Lévy observed was that the plane can be tiled by copies of the Lévy dragon. This means that there is a sequence of sets congruent to the Lévy dragon that are non-overlapping and whose unions is the entire plane. Click for more details.

Another property shown by Lévy is that the dragon has non-empty interior [see Edgar]. The paper by Duvall and Keesling investigates some properties of this interior. In addition, the authors develop a theoretical approach to computing the Hausdorff dimension of the topological boundary of attractors of iterated function systems, and apply this theory to estimate that the boundary of the Lévy dragon has Hausdorff dimension of approximately 1.954776399.

Lévy to
Heighway
Animation
(106Kb)

The construction of the Lévy dragon and the Heighway dragon are very similar. In each case one can start with an isoceles right triangle and replace this triangle with two isoceles right triangles so that the hypotenuse of each new triangle lies on one of the equal sides of the old triangle. The difference is how those new triangles are placed relative to sides of the old triangle. For the Lévy dragon, both are placed towards the "outside"; for the Heighway dragon, one is placed pointing out while the other is placed pointing in. Because of this similarity, it is perhaps not surprising that one can transform the Lévy dragon into the Heighway dragon through a continuous transformation. Indeed, for each t in the interval [0,1] define an iterated function system by

Let A(t) be the unique attractor of the iterated function system corresponding to the value t. Then A is a continuous function from the unit interval to the space of compact sets with the Hausdorff topology, with A(0) equal to the Lévy dragon and A(1) equal to the Heighway dragon. The Quicktime movie gives an animation of A(t) as t goes from 0 to 1.

Lévy, P. "Les courbes planes ou gauches et les surfaces composée de parties semblales au tout," Journal de l'École Polytechnique (1938), 227-247, 249-291.
Lévy, P. "Plane or space curves and surfaces consisting of parts similar to the whole," in Classics on Fractals, Gerald A. Edgar, Editor, Addison-Wesley, 181-239.
Duvall, P. and J. Keesling. "The Hausdorff dimensions of the boundary of the Lévy dragon," Topology Atlas Preprint #209 (1997), http://at.yorku.ca/p/a/a/h/08.htm.

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