SimuLab 14: The Fractal Dimension of Hele-Shaw Patterns

As already discussed in Unit 2.2, a line has length L = L¹. A square of side L has an area equal to L². A cube of side L has volume L³. In each case the exponent of L reveals the dimension D of the object. A fractal can have a fractional value of D.

1. During your experiment, for each volume V of fluid injected you obtained the radius R of the pattern and we plotted the results on the linear graph of Figure 4.12. If the fingering pattern is a fractal then we expect the relationship.

where c is a constant of proportionality and D is the fractal dimension. If we take the logarithm of both sides of 3) we obtain

It follows that if we plot logV versus logR and obtain a straight line, then the slope of this line is D, the fractal dimension, and the intercept is logc.


Using either a computer graphing program or log-log graph paper, plot logV versus logR and determine D from the slope.


2. You should also determine the fractal dimension of the final pattern by capturing the image using a digital camera, scanner, or a video camera if the computer has video capabilities. After the image has been converted into a digital image, use the Fractal Dimension program to determine the value of the dimension D.