# Aggregation: Growing Fractal Structures

You have probably observed before that no two snowflakes or two lightning bolts are ever exactly the same. Though two such patterns may have the same overall shape, if you look closely you will see that they differ in the details of their structure. The same is true for other natural fractals as well. Each example is unique.

Now we make another observation. If two fractals are of the same class (e.g., two bolts of lightning), they will be somehow similar in appearance to each other.

The question is: What is the basis of that similarity? How can we describe, classify, and measure different random fractal patterns in nature? That is the subject of this chapter and the activities described on the following pages.

What Do You Think?
In THE IMAGE GALLERY there are several pictures of fractal structures.
• What do they have in common? How are these objects similar to each other in appearance? In what ways do they look different?

• There are many ways to rank objects. For example, you can line up students according to weight. Or you can line them up by height, shoe size, age, or hair color. Look at the photographs and figure out different ways to line up these images in order. Line them up according to as many different criteria as you can think of. How inventive can you be?

• The pictures show very different objects, yet their images look rather similar in many ways. Why do you think these objects look so much alike? What are possible ways that their similarity might arise from a common cause?

Is a coastline a line? Not really, because a line is one-dimensional, whereas a coastline is a random fractal that has a dimension whose value is between 1 and 2. Other random fractal patterns also have dimensions between 1 and 2: a snowflake, a nerve cell, a lightning stroke. The growth of these structures can be modeled by a process called aggregation, in which random walkers dance around the growing structure and stick to it when they touch it. The resulting jagged pattern is called an aggregate. Depending on the details of the growth process, some aggregates are natural fractals, others are not.

We have put online several different aggregation experiments (here) as well as described growing a metal crystalline deposit in our MUSEUM EXHIBIT.

How can we describe the process by which a pattern grows (aggregates)? Can we mimic the way a charged atom (ion) in a solution dances around, then deposits on the central electrode (becomes an uncharged metal atom)?

What does it mean for an ion ``to dance?'' Dancing means to stagger around randomly. A dancing ion is taking a random walk! We can use our understanding of random walks to mimic the process of electrochemical deposition.

Another word for mimic is model. We model the aggregation process using our knowledge of a random walker who staggers around and sticks to a growing structure. Do you think that the electrodeposition experiment can be described using this model?

Compare this model to your picture of the motion of ions in the electrodeposition experiment. What component of the experiment does the walker represent? What component does the black pattern represent? Who or what is doing the ``die throwing'' in the electrodeposition experiment? In what way do the ions in the electrodeposition experiment move differently than the walker in the aggregation model?

This applet explores this model. It is called Diffusion-Limited Aggregration.

Now that you've had a chance to experiment, can you answer these questions?

• You have just discovered how a random walker can be used to grow spidery patterns similar to those found in nature. Does this mean that random processes are actually involved in forming these objects in nature? Or are lightning, nerve cells, termite tunnels, and electrodeposition patterns formed by totally different processes, processes that have nothing to do with random walkers?

• The discovery that a random walker can be used to grow spidery patterns similar to structures found in nature is very important to scientists. Why do you think this discovery is important? In what ways could this discovery prove to be useful?

• Select Straight under Particle Movement. In the ``straight'' setting, each particle moves in a straight line but in a direction that is chosen randomly. How is the movement of the particle different compared with the Random setting? Does the pattern grow in more steps or fewer steps for the Straight setting than for the Random setting? Is the pattern more leggy or more compact for the Straight setting than for the Random setting?

This lesson is taken from Fractals in Science. The JAVA applet was written by Anna Umansky based on a program by Sergey Buldyrev. Copyright 1996-2000, Center for Polymer Studies.