Famous Fractals

Koch Snowflake

Koch Snowflake is one of the most famous fractals. It is a base-motif fractal with the following base and motif:

During the first iteration, we substitute every side of the triangle with the motif:

During the second iteration, we substitute every one of the 12 line segments with the same picture:

We continue doing the same infinitely to get the Koch Snowflake:

FRACTAL DIMENSION

We can find the fractal dimension of this fractal using the similarity method. Since in the motif, there are 4 identical line segments that are each 1/3 long, the dimension is log 4 / log 3, which is approximately 1.26.

LENGTH

At every iteration, this fractal becomes 4/3 times longer. After an infinite number of iterations this fractal, just like all fractal curves, will become infinitely long.

AREA

Suppose the area of the original triangle is 1. Then, after the first iteration the total area of the three triangles added is 1/3 and the area becomes 4/3:

After the second iteration, the total area of the 12 triangles added is 4/27, and after the third iteration, the total area of the 36 triangles added is 4/81. After that the areas added become 3 times smaller with every iteration. To calculate the sum of all areas added, we use the formula for the sum of an infinite geometric sequence, S = a / (1–r), where a is the first number of the sequence and r is the common ratio. In our case, the sum would be (4/27) / (1–1/3) = 2/9. If you add this to the area of 4/3 that we had after the first iteration, the area will become 14/9. That is the area of the entire snowflake.

The Koch Snowflake can be very beautifully filled in by any of the series of fractals called the Snowflake Sweeps.