Famous Fractals   

Koch Antisnowflake

Koch Antisnowflake is a base-motif fractal formed with the base and motif below. Its difference from the Koch Snowflake is that the motif is placed inside the triangle instead of the outside.

We start with a triangle and substitute every side with the motif:

We continue substituting every line segment with the motif:

After repeating the substitutions an infinite number of times, we get the Koch Antisnowflake:

FRACTAL DIMENSION

The motif of the Koch Antisnowflake contains 4 identical line segments, the length of each of which is 1/3. Using the similarity method, we can find the fractal dimension to be 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

In the calculations of the area of the Koch Snowflake, we found the areas of the triangles added to the figure. They turned out to be 1/3 for the first iteration, and 2/9 in total for all of the other iterations. For the Koch Snowflake, we added these numbers to the area of the original triangle, which was 1. However, in the Koch Antisnowflake, the triangles face inside the figure and take away some area, so the areas have to be subtracted from 1. Subtracting 1/3 and 2/9 from 1 we get 4/9, which is the area of the Koch Antisnowflake.