Famous Fractals

Polya’s Sweep

Polya’s Sweep is a base-motif fractal formed with a very simple base and motif:

To form this fractal, you substitute the original line segment with the motif. In all of the next iterations, however, you alternate between the original and the flipped version of the motif, since the fractal is a sweep:

As you can see, you also alternate the version of the motif that you are using for the first line segment. If you continue the iteration infinitely, you will get a right isosceles triangle with a leg of length 1 at the end. This means that Polya’s Sweep is also a Peano Curve.

DIMENSION

Since Polya’s Sweep 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 Polya’s Sweep becomes sqr(2) times longer. Just like all fractal curves, the length of this fractal is infinite.