Larry Riddle, Agnes Scott College

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

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

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.

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.

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.