## 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.

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:

- divide a line segment into three equal parts
- remove the middle segment (= 1/3 of the original line segment)
- replace the middle segment with two segments of the same length (= 1/3 the original line segment) such that they all connect (i.e. 3 connecting segments of length 1/3 become 4 connecting segments of length 1/3.)

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.

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).

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