Manual Box-Counting Directions

The box counting dimension of a shape in the plane can be estimated by the slope of the plot of Log(number of boxes to cover the shape) vs Log(1/box side length).

1. Place the transparency (smooth side up) with the largest grid scale on top of the curve so that it is completely covered by the boxes. To prevent movement, secure the transparency to the picture with tape.

2. Count the number of boxes that contain any portion of the curve. Record the number on the data sheet.

3. Repeat the next two transparencies. Assuming the largest grid has boxes of side length 1, the next two have boxes of side lengths 1/2 and 1/4.

A transparency with boxes of side length 1/8 is provided. Including this will give more accurate results, but it need not be used to illustrate the mechanics of box-counting.

4. Compute log(number of boxes) and log(1/box side length) for the data gathered.

5. Find the slope of the line of best fit through the points determined in step 4. This can be done in any of three ways.

a. Plot the log data by hand on a sheet of graph paper, draw the best estimate of the line connecting the data points, and estimate the slope of this line.

b. If the data are stored in a graphing calculator, use the calculator to find the linear regression equation that best fits the log data. Record the slope of the line.

c. If the data are entered in a spreadsheet, use the chart or linear regression capabilities of the spreadsheet to graph and find the slope of the line of best fit.

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