Types of Fractals

Dusts and Clusters

WHAT ARE THEY?

Take any shape, a line segment for simplicity. Call it the base. Cut out some parts from it. What you should get is several smaller line segments. We will call these line segments the motif. Now take each of them and cut out the exact same parts that you cut out before. After continuing this process, you get an infinite amount of points, which form a fractal called a dust. What you did is called generator iteration, the same process that is used for base-motif fractals. In fact, a dust is a base-motif fractal in which the motif is the same as the base, only with some parts cut out. The most famous dust is most simple one, called the Cantor Set, in which you divide a segment into three equal parts and cut out the middle third:

If you are familiar with similarity method, you can probably realize that the fractal dimension of all dusts is less than 1.

2D CLUSTERS

Instead of taking a line segment as a base, you can take a 2-dimensional figure. This will give you a fractal called a cluster. For example, using a square lets you create fractals such as the Sierpinski Carpet:

If you use an equilateral triangle, you can create fractals such as the Sierpinski Triangle:

Clusters are very useful in modeling the galaxies, which themselves tend to cluster. Clustering was also found in chromatin.

AND 3D, OF COURSE

You can also use a 3-dimensional figure such as a cube or a pyramid for the base. This lets you create fractals such as the Sierpinski Pyramid and the Megner Sponge: