Famous Fractals

Cesaro’s Sweep

Cesaro’s Sweep is a base-motif fractal with the following simple base and motif:

To form it using generator iteration, we take the base and substitute it with the motif. However, since it’s a sweep, we flip the motif over at the next iteration:

If we continue an infinite number of times, we will get a right isosceles triangle, the size of who’s hypotenuse is 1:

Since this fractal fills a 2-dimensional figure, it can be considered a Peano curve.

DIMENSION

Since Cesaro’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 Cesaro’s Sweep becomes sqr(2) times longer. Just like all fractal curves, the length of this fractal is infinite.
An interesting fractal can be formed if we let the angle in the motif be a little greater than 90 degrees, let’s say 95 degrees. If we use a square for the base, we get the following picture: