HandsOn 5 - Covering a Coastline with Boxes

Thus far we have used the ruler method for measuring the dimension of a coastline. An alternative method is called the grid method or box method or covering method. The grid method is a bit more versatile than the ruler method, and can be used for different kinds of fractals. Here is how it works:

1. Get out the map whose coastline you measured originally.

2. Take a second piece of blank paper approximately the same size and cut out a square that just covers the entire coastline to be measured. Call the edge-length of this square L = 16.

3. Now fold the covering square into fourths and cut along the fold-lines. Each of the four squares has an edge length L = 8.

4. Predict: How many of these L = 8 squares will it take to cover the coastline? The word "cover'' means "cover up'': the squares lie along the coastline without overlapping, but so that no piece of the coastline is visible. Enter your prediction in a copy of Table 2.2.

5. Now cover the coastline with L = 8 squares and enter the result in your copy of the table. How good was your prediction?

6. Fold each of the smaller covering squares into equal fourths and cut along the fold lines. Predict how many of these squares of length L = 4 it will take to cover the coastline. Enter your prediction in your copy of Table 2.2

7. Now cover the coastline with squares of L = 4 and enter the result in your copy of Table 2.2

8. Repeat this process with squares of edge-length 2, and 1. In each case predict the number of squares that will be needed to cover the coastline, and enter the result in your copy of the table before measuring and entering the actual number of squares needed to cover the coastline. Were your predictions closer to your measured values this time than when you predicted results for the ruler method?

10. Get out your original log-log graph of the coastline results. Plot the data from your copy of Table on the same graph. This time the vertical scale will be "number of squares to cover coastline'' and the horizontal scale will be "edge length of square.'' Is the result approximately a straight line? If so, is the slope of the line the same as you found using the ruler method?

Table 2.2: Covering the Coastline with Boxes (used in HandsOn 1 and SimuLab 1)
 Edge length of PREDICT: MEASURE: Square Number of Squares to cover Number of Squares to cover Coastline Coastline 16 8 4 2 1

 Q2.10: Based on your use of the ruler and grid methods, which do you find more efficient? Does it make sense to ask which method is more "accurate''? Explain.

Next: SimuLab 1 - "Walking" Along a Coastline