Tutorial Chapter 3
Brownian Self-SimilarityWHY DO WE NEED IT?
In the previous lesson you were introduced to self-similar fractals. Letís look at one of them, called the Koch Snowflake.
No matter how far you magnify this fractal, you will see a fragmented picture. Compared to a straight line, the Koch Snowflake is obviously better in describing a natural shape such as a coastline or a river. However, there is a major drawback to that. Have you ever seen such a perfectly symmetrical shape in an atlas? Obviously, normally self-similar fractals are too regular to be realistic. To make fractals more realistic, we use a different type of self-similarity called Brownian, or statistical self-similarity.
AND WHAT EXACTLY IS IT?
In 1828, Robert Brown was studying the motion of microscopic particles. What he discovered was named Brownian motion after him. If you plot the location of some particle at certain intervals of time, you will get a fragmented trajectory with lines randomly located in space.
Now, take one of these lines and plot locations at smaller intervals of time. What you will get is a smaller fragmented line made up of randomly located parts.
If you take one of these lines, you will find that it is made up of smaller lines as well. If you read the previous lesson, this should remind you of self-similarity. However, this self-similarity is different. Although each line is composed of smaller lines, the lines are random instead of being fixed.
Brownian self-similarity is found in plasma fractals. They are very useful in creating realistic coastlines and landscapes. In addition, the same concept was used to create a realistic model of the economy.