• Objectives
• What are Fractals?
• Constructing Simple Fractal Images
• Fractals and Nature
• Scaling and Self-Similarity
• Fractal Dimension
• Further Reading and Activities

Objectives

The objectives of this brief presentation and related activities are the following:

• View and analyze fractal images and patterns
• Create simple fractal images
• Explore fractals in nature
• Study scaling and self-similarity
• Learn about fractal dimension

What are Fractals?

The subject of fractals was invented by Benoit Mandelbrot. He coined the word fractal to refer to an object that had a fractional dimension. An image of the famous Mandelbrot set is shown here.

There are many books and web sites devoted to the subject of fractals and fractal images. A good place to start learning about this exciting subject is the web page on Fractals in The Math Archives.

Let us start this brief introduction by looking at some examples of fractal images and patterns. We shall check out the following web sites: 

• fractalarts.com
• Spanky Fractal Database

• Fractals possess the property of self-similarity. They can be subdivided into smaller parts, each of which is similar to the original. 
• Fractals can be created by a process of iteration -- that is by repeatedly drawing the same pattern, usually on a decreasing scale.
• Fractals can have fractional dimension.

Fractal images may be constructed by a process of iteration. Let us try to draw fractal images of the simple objects listed below. You will be able to see the sequences of steps that are used in creating these images by clicking on their names.

1. Trees
2. Koch Snowflake
3. Sierpinski Triangle
4. Apollo - fractal design with circles
5. Cantor Dust

More complicated fractal images can be created using computer software that are freely available through the web or can be purchased as commercial packages.
 

Fractals and Nature
 

Nature is filled with a large number of different types of fractals. Some examples are: Trees, Plants and Vegetables Snowflakes Landscapes, Coastlines and Clouds Parts of the Human Body Weather Radio Signals (e.g. Cantor Dust)

This photograph of a tree is an example of a fractal object in nature. It comes from an extensive collection of photographs by Professor J. C. Sprott of the University of Wisconsin-Madison.
 

Scaling and Self-Similarity

In simple language we may say that two geometrical objects are similar if one is an enlargement of the other. For example, two triangles will be similar if their corresponding angles are equal to each other and the corresponding sides are magnified by the same factor. The following figure shows two similar triangles:

Notice that the smaller triangle is scaled by a factor of two to obtain the larger triangle.
 

The above example displays the similarity of  two different objects. Now, look at the Sierpinski triangle in the picture on the right. If we take any of the smaller triangles with a white background, we can enlarge it to fit the large triangle and have the same appearance. This is self-similarity, which is a characteristic the property of fractals.

Fractal images can be subdivided into smaller parts that are similar to the original. That is, if we enlarge one of these smaller parts it would look like the original.

An interesting activity that you might consider is to look at images occurring in nature and see whether they contain features of self-similarity.
 

Fractal Dimension

We know that the real world is three-dimensional. That is because at any point in space we have three independent directions. These are north-south, east-west and up-down. Similarly, a thin line is one-dimensional and a flat surface (plane) is two-dimensional. Notice that these dimensions are integers. In the case of a fractal object, the dimension is generally not an integer. Although the formal mathematical definition of fractal dimension is beyond the scope of our present discussion, we shall take a simple approach to see how a fractal could have a fractional dimension.

First consider a line of length 1. Let us scale it by a factor of 2. The new image contains two copies of the original image as seen in the following figure:

Let us take N = number of copies of the original image in the new image and let S = scale factor. The dimension, d, of an object may be expressed in a simple way by the equation N = S^d. 

For the line, we have N = 2 and S = 2. Putting these in the equation for the dimension, we get 2 = 2^d. This gives d = 1, which is the dimension of the line.
 

Next, let us take a rectangle and scale it by a factor of two. The new rectangle contains four copies of the original. We have N = 4 and S = 2. The dimension equation reads 4 = 2d. We get d = 2, which is the dimension of the plane.

 For a three-dimensional object let us take a cube with each edge having length equal to 1 and then enlarge it by a scale factor of 2. The new cube will contain 8 copies of the original cube. That is, N = 8 and S = 2. The dimension equation reads 8 = 2^d, which gives d=3.
 

Now, look at the Sierpinski triangle. If we take the triangle with the blue background and enlarge it by a scale factor 2, we get the large triangle. There are 3 copies of the blue triangle in the large triangle. So,  N = 3 and S = 2. The dimension equation is  3 = 2d . Notice that the value of d must lie between 1 and 2.

To get the exact value of d we must use logarithms and a calculator. If you are familiar with logarithms you will know that we can write d = (log 3)/(log 2). A calculator gives this value to be 1.58496... , which is not a whole number.

In the case of any fractal object, we can use logarithms to write the dimension as d = (log N)/(log S).

Finally, let us consider the Koch Snowflake. What is the dimension of its boundary and what happens to the length of the boundary as we proceed with the iteration? The snowflake is created by removing the middle third of a side and inserting, in its place, two sides of an equilateral triangle. The following picture shows the first two iterations along one side AB of the original triangle.

The boundary consists of line segments. When we go from one stage to the next, each line segment is replaced by four line segments, each of which is one-third the length of the larger line. So, N = 4 and S = 3. Therefore, 4 = 3^d or we can write d = (log 4)/(log 3), which is approximately 1.2618.

To compute the length of the boundary, we shall focus on the portion between the vertices A and B of the original triangle. Suppose that the initial length of AB is 1. After the first iteration, it becomes 4 times 1/3, which is 4/3. After the second iteration, the length is 16/9, which is the same as (4/3)². That is, at each step, the length between A and B is enlarged by a factor of 4/3. After m iterations, the length will be (4/3)^m. This number approaches infinity as the number of iterations m approaches infinity. Although the length of the boundary becomes infinitely long, the area inside remains finite because we can enclose the figure within a finite circle.
 

Further Reading and Activities

• For further reading, there is an excellent book by James Gleick entitled

Chaos : making a new science . This was a best-seller at one time.
• You could look up the materials in the web sites that are listed above and visit other sites that are mentioned in those web pages.
• An interesting question that you will come across in your reading is what is the length of the coastline of Britain? You will find that the answer depends on the length of the measuring stick that one uses. This is related to the problem of the length of the boundary of the Koch Snowflake, which we discussed above.