Chaos & Fractals


Simply put, a fractal is a geometric object that is similar to itself on all scales. If you zoom in on a fractal object it will look similar or exactly like the original shape. This property is called self-similarity. An example of a self-similar object is the Sierpenski triangle show below.

Sierpinski Triangle

As one looks closer we observe that the large triangle is composed of three smaller triangles half the size (side length) of the original, which in turn are composed of three smaller triangles, and so on, and so on. On all scales the Sierpenski triangle is an exactly self-similar object.

The property of self-similarity or scaling is closely related to the notion of dimension. In fact, the name "fractal" comes from property that fractal objects have fractional dimension.

A one dimensional line segment has a scaling property similar to that of fractals. If you divide a line segment into N identical parts, each part will be scaled down by the ratio r = 1/N (e.g. cut a line in two equal pieces and you have two lines each of half the original length). Similarly, a two dimensional object, such as a square, can be divided into N self-similar parts, each part being scaled down by the factor r = 1/N(1/2) (i.e. if you cut a square into 4 equally-sized squares, then each new square is half the size (side length) of the original square).

The concept of self-similarity naturally leads to the generalization to fractional dimension. If one divides a self-similar D-dimensional object into N smaller copies of itself, each copy will be scaled down by a factor r, where

r = 1 / N(1/D)

Now, given a self-similar object of N parts scaled down by the factor r, we can compute its fractal dimension (also called similarity dimension) from the above equation as

D = log (N) / log (1/r)

As an example, let us compute the dimension of the famous curve of Von Koch, which is sometimes referred to as the "Koch Snowflake." The Koch Snowflake is generated by a simple recursive geometric procedure:

To complete the shape, the above procedure is repeated indefinitely on each line segment on the side of a triangle. Images showing the procedure and complete Koch Snowflake are shown below.

Von Koch Snowflake

Von Koch Snowflake

Its fractal dimension is given from the definition of the curve: N = 4 and r = 1/3 (remember 4 segments each 1/3 size of the original line segment).

Dimension = log (4) / log (3) = 1.26

Another interesting property of the Koch Snowflake is that it encloses a finite area with an infinite perimeter.