Imagine a forest in which the trees were planted by scattering seeds randomly all over the ground. The trees are now full-grown, tall and green. Some trees stand alone; others are in clusters. The forest is peaceful on a quiet, windless day.
Now imagine that all the trees along one edge of the forest are suddenly set on fire. As each tree burns, the fire spreads rapidly to the trees next to it. Every tree in the same cluster burns, as the fire spreads from each tree to its closest neighbors. But fire does not spread from one cluster to another, because it cannot jump across the space that separates clusters; there is no wind.
If the trees are planted sparsely, with just a few trees scattered over a large area, chances are the fire will burn only a small number of trees and then die out; most of the forest will be safe. That's because sparsely planted trees, scattered randomly, tend to stand alone or in small clusters separated from one another.
In contrast, suppose that the trees are planted very densely, with lots of trees crowded into a small area. Then the fire is likely to spread through nearly the entire forest. That's because densely planted trees, scattered randomly, tend to form large clusters. There may even be one large cluster that reaches from one side of the forest to the other. The trees in such a large cluster form a connected path that allows the fire to spread from tree to tree across the entire forest.
A cluster that reaches from one side of the forest to the other is said to percolate, and the model we have just described for a burning forest is called percolation. We say a cluster percolates when it forms a connected path from one side of the forest to the other. The cluster that spans the forest is called the ``percolating cluster.'' (A ``percolator'' is a device for making coffee. In a coffee percolator, water percolates--- finds a connected path---through the narrow spaces in a thick layer of ground coffee.)
Think again about the burning forest. When tree density is low, trees do not form a percolating cluster, and the fire does not spread very far. In contrast, when the density is high, trees do form a percolating cluster, and the fire spreads across the forest. But what happens in between? And what does a percolating cluster look like, anyway? Can a percolating cluster be a fractal?
To answer these questions, we study the method of growing a ``forest'' by hand. We call this method a model. Then we study the same model using the computer.
Work in pairs. You need:
First create a random forest on a square grid, with 25 cells in the grid.
Now the model forest is grown. Does it look to you like a real forest? How is it different?
We are now ready to start a fire in our model forest, and see how far it spreads. The fire starts along the left edge of the forest and spreads from one tree to another if they are neighbors (up, down, right, left) on the grid. Here's how it works:
Now you have burned your forest. Did the fire percolate? That is, did it spread all the way from the left edge to at least one tree on the right edge? Or did it stop part way across the grid?
Now you have seen what happens in a model forest where about 50% of the squares are occupied by trees. What would happen if there were more trees? Would the fire be more or less likely to spread all the way across?
Grow a new forest on the same size grid, but this time roll a die for each square, placing a tree if the die comes up 1, 2, 3 or 4 and leaving the square blank if the die shows 5 or 6. Now predict what you expect to happen when you start a fire along the left edge. Will more or fewer fires burn across to the right side of the forest than before? Try it!
Now play the Blaze Applet, the computer game of forest management and fire fighting. Here is how the game works. You are responsible for growing and harvesting trees on a plot of land. The more trees you harvest, the higher your income. Unfortunately, your forest is located in a high-risk fire region; when the forest is grown, a blaze begins along the left edge. The more trees survive the fire, the more your profit.
The applet program allows you to select the density of tree growth,
the probability, labeled
p, that a tree will grow on each site. In the
preceding activity, you flipped a coin, so the probability was one half,
or p=0.5. You can select the value of
p between 0 (no
trees) and 1 (a tree on every site). You can also select the dryness of
the forest, which determines how fast the fire spreads.
What value of
p, the tree probability, should you use? If you plant
trees at low probability, trees will usually be separated from one
another (low density), so the fire will not spread. As a result, not
many trees will burn, but you won't harvest many trees either, since
there are not many trees altogether. On the other hand, if you use a
high tree probability, the trees will typically be next to one another
(high density), fire will spread across the forest, and again you won't
have much of a harvest. The trick is to find a particular tree density
(a particular probability
p) that gives you the chance for a large
harvest but allows you to control the spread of fire.
In addition to controlling tree density, you can fight the fire by dumping water on individual trees. Your job is to maneuver your helicopter and drop water to stop the spread of the fire. The helicopter moves to the position where you place the cursor and drops water when you click the mouse. Any tree that you dump water on does not burn. The idea is to wet trees ahead of the fire to stop it from spreading. You have a limited amount of water to dump on the forest.
When the fire has burned out, your score is shown in the lower right of the screen. The score depends on the number of unburned trees and the dryness of the forest.
Now that you've had a chance to experiment, can you answer these questions?
p at which you can keep the fire
from reaching the opposite side (right side) of the forest, no matter
how effective your strategy? Do you ``run up against a wall'' at some
p, finding it impossible to keep the fire from crossing the
forest for higher values of