Famous Fractals

Dragon Curve

The Dragon Curve can be formed by either folding paper or by using generator iteration.

FOLDING PAPER

Consider taking a strip of paper in half, and then unfolding it do that the angle formed is 90 degrees. Then consider folding it twice and three times:

Now consider folding it an infinite number of times and then unfolding it:

Since this fractal fills a 2-dimensional figure, it is a Peano curve.

GENERATOR ITERATION

The Dragon Curve is a base-motif fractal and can also be formed using generator iteration with the following base and motif:

We take a base and substitute it with the motif. However, since this fractal is a sweep, we use a flipped version of the motif for the first and every other odd-numbered line segment.

FRACTAL DIMENSION

Since the Dragon Curve is a Peano curve, its fractal dimension is 2. We can also find it using similarity method. Since the motif has 2 identical line segments of sizes 0.5·sqr(2), the dimension is log 2 / log (2·sqr(2)) = 2.

LENGTH

With every iteration, the length of the Dragon Curve becomes sqr(2) times longer. Just like all fractal curves, the length of this fractal is infinite.

TWINDRAGON

The Twindragon can be formed by putting two Dragon Curves on top of each other:

As you can see, this figure is itself a fractal since it has clear self-similarity. The "skin" of the Twindragon is also a base-motif fractal.