Curves are measured by length, planar figures by area, or length squared, solid objects by volume, or length cubed. In more general terms Equation 3. In the case of fractals, we want to know what the dimension is. We can find it from Equation 1 by taking logarithms and solving for Dimension. For our fractal figure, each step in the construction replaces 3 line segments by 4.
Fractals have fractional dimensions.
One way to study such natural fractals is to measure them at various scales and analyze the results. For example, Richardson Analysis involves measuring a landscape feature with steps of different sizes. The smaller the step, the greater the total length as smaller details are measured. We can plot the above data on a log-log plot. The points fall on or near a straight line, whose slope is m.