SimuLab 10: Measuring Fractal Dimension
We learned with the Coastline program (SimuLab 1)that a computer can quickly cover a pattern with boxes of various sizes, count the number of boxes, and plot the result in order to measure the dimension of that pattern. In this section we learn to use a new and even more powerful computer program called Fractal Dimension (see Java Applet Page) to analyze patterns grown in the laboratory, as well as patterns simulated by computer programs such as Coastline.
2. Select one of the images from the Sample Images menu.
3. Click on the red square at the left of the Toolbar. This covers the image with little boxes.
4. Click on the size control boxes at the top of the window to change the box size, then click again on the red square in the toolbar. The machine covers the coastline with boxes of the new size.
5. Change the box size and repeat the count with several box sizes.
6. Click on the graph icon, fifth from the left in the toolbar. This brings a log-log plot to the front. This plot gives the value of the slope, which is equal to the dimension D. If you wish to fill in the graph with more dots, go back and do additional counts with boxes of appropriate size.
7. Click on the table icon, sixth from the left in the toolbar. This opens a data table listing box lengths and counts. If you feel that one or more of the data points on the graph should be eliminated from the slope measurement, click on the number in the # Boxes column. This will "gray out'' that row and the dot for that entry will disappear from the graph. (Restore the data point by clicking again on that item in the table.)
8. Analyze other images from the Images menu using the box method. In particular, you may want to check that a straight line (whether horizontal or diagonal) has a dimension of one. Is a solid square 2-dimensional, according to the Fractal Dimension program? You will also be interested in the Koch curve from the Images menu. Is the fractal dimension of the Koch curve determined by this program the same as your measurement in HandsOn 8 and the theoretical result of Eq. 4.2?
Measuring the Pattern from the Electrochemical Deposition Experiment: Now call up the image of the pattern you grew in the electrodeposition experiment and measure its dimension using the box method. How does this value compare with the value of the dimension you obtained using the current and radius values obtained during the experiment?