Ten leaves each of three different trees were scanned. The fractal dimensions of the perimeters of the resulting images were measured using the program Fractal Dimension. Leaf perimeters were found to have fractal dimensions near 1.0. Analysis showed that the fractal dimensions of oak and beech leaves were distinguishable, but the fractal dimension of chestnut leaves was not distinguishable from either of the other two. In general, fractal dimensions alone can not be used to classify leaves by genus.
Leaves from the beech and oak trees were statistically distinguishable (t = 5.52), but leaves from beech and chestnut, and from oak and chestnut were not (t= 1.15; t = 0.069). (The measurement of t compares the differences between means, taking into account the variation in data due to measurement uncertainty; when t > 2, we can be 95% confident that the difference between means is genuine, and not a result of measurement error.)
In general, trees can not be distinguished by genus using the fractal dimensions of leaves. Chestnut leaves could not be distinguished from either beech or oak leaves. However, some differences in leaf shapes do show up in their fractal dimensions. Beech leaves have a significantly higher fractal dimension than oak leaves, presumably due to the amount of serration in their edges.
Another question was whether leaf perimeters have a fractal dimension significantly different from that of a straight line, 1.0. Our fractal graphs consistently yielded straight lines, with no outlying data points. Although the values were close to 1.0, it seemed clear that beech and oak leaves have fractal dimensions greater than 1.0. To check whether this might have been an artifact of the Fractal Dimension program, we used the program to measure the fractal dimension of a straight line. However, the program failed to yield reasonable graphs; there were always outlying points (due to the way the program placed boxes over the line), leading to wide variation in fractal dimension measurements.
We suspect the program is not designed to measure fractal dimensions for straight lines, but we do not know whether this limitation compromises the reliability of the data for the fractal dimensions of leaves.