Agnes Scott College
Larry Riddle, Agnes Scott College

Tiling the Plane with Lévy Dragon

In his 1938 paper, Lévy wrote [see Edgar]

Moreover, it has the very curious property that, if [the dragon curve] were materialized, one could tile the plane with it, i.e., one can cover the plane with identical copies of this curve such that no two copies intersect, and such that no space is left uncovered.

Here is one way to visualize why this tiling works. The plane can certainly by tiled by unit squares. Divide each unit square into four congruent isosceles right triangles, each a congruent copy of L0, the isosceles right triangle with base along the unit interval and vertex at (1/2, 1/2). This was the triangle used in the description of of the construction of the Levy dragon. Applying the first step of the Lévy dragon construction to each of these triangles produces four congruent copies of L1 that still cover the unit square (see Figure 1).

Figure 1
Figure 1

Now apply the next step of the construction to each of these regions to get four congruent copies of L2. Some of the new triangles will now point outward, leaving what appears to be empty space in the square in the third diagram in Figure 1. But of course, we are also doing this construction to each of the surrounding unit squares, and some of the triangles from those surrounding squares produce new triangles that fill in the gaps. Try it with the following animation of steps using copies of L2 from the 8 surrounding squares to produce the following tiling.

 


Tiling
Animation

L2tiling

L2unitsquare Notice in the tilings using L2 that the unit square (shown to the right) can be split into 4 smaller squares of area 1/4 that resemble the original pattern of four isosceles right triangles in the tiling with L0. We saw how the plane continues to be tiled in going from L0 to L1 to L2. Because the tiling for L2 consists of the same pattern as for L0, only on a smaller scale, the same construction would show that L3 and L4 continue to tile the plane.


 
 
 
 
 

Here is a picture of L4. Click on the buttons to the left to see congruent copies of L4 starting to tile the plane. Again, the gaps you see would be filled in from the copies generated by the unit squares further outside.

L4


Quicktime
Tiling
Animation
Now the tiling for L4 can again be split into squares of area 1/16 that resemble the original pattern of four isosceles right triangles, so we can proceed just as before. Continuing this way, an induction argument can then be used to show that congruent copies of Lk can tile the plane for every value of k. This result will therefore remain true for the limit set, and so Lévy's dragon will tile the plane. Below is an image with four copies. This picture is a bit misleading because it was drawn on a computer screen where the pixels, although small, still have positive area and thus the copies appear to overlap. In the limit, however, this would not happen.

LevyTiling4copies

Reference

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.