Levy Curve is a base-motif fractal which is formed with a line segment for the base and one of the simplest motifs:
To generate it, you take a line segment and substitute it with the motif. You then take the figure and substitute each of its 2 segments with the motif:
As you continue generating the Levy Curve, the shape becomes amazingly more and more complicated and after a large number of iterations forms a very complex fractal. Although it might not be easy to see at first, the shape is perfectly self-similar, as you can see in the final picture below:
If you use the same motif as the Levy Curve, but with a square for the base, we can get the Levy Tapestry.
You can find the fractal dimension of the Levy Curve very easily using the similarity method. There are 2 identical line segments in the motif, each of which has to be magnified sqr(2) times to become the same size as the base line segment. The dimension is, thus, log 2 / log(sqr(2)), which is equal to 2. This means that the Levy Curve is a Peano curve.
At every iteration, the fractal becomes sqr(2) times longer. This means that after an infinite number of iterations the fractal will become infinitely long, just like all fractal curves do.