Fractal Dimension of Leaves

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Abstract

The purpose of this lab is to determine the dimensions of six different leaves. The dimensions that were obtained ranged from 1.7 to 1.9.

Introduction

Dimension can be defined as the number of parameters or coordinates that is used for describing mathematical objects. For example, lines are defined as one dimensional, while squares are two dimensional. Fractal are irregularly shaped objects formed of repeating similar patterns. They cannot be described with a whole number dimension. Objects such as the outline of leaves, trees, and coastlines are fractal in nature. The fractals of the outline of leaves are between one and two dimensional. The fractals of leaves is a repeatedly dented or deformed circle.

Procedure

Six different types of leaves were scanned into a computer and were saved as black and white bitmap files. A program called Fractal Dimensions was obtained from Boston University's Polymer Studies website. Fractal Dimensions was then used to analyze each leaf image. The program calculated the number of boxes of a certain size needed to fill the space around an image. This process was applied to each leaf. Beginning with an initial size of two, the box size was slowly increased by intervals of two until twenty was reached. The program then displays a log-log graph of the box size versus number. Fractal Dimension then found the slope of the best fit straight line, which is the fractal dimension of the leaf.

Data

Box Size Number of Boxes
Juniper Maple Jr. Maple Leaf 4 Leaf 5 Leaf 8
2 51223 208667 494931 52990 28266 41948
4 13784 52869 124496 13659 7416 10669
6 6483 23750 55927 6332 3442 4894
8 3943 13529 31550 3667 1952 2754
10 2684 8836 20320 2420 1336 1866
12 1889 6132 14163 1756 954 1322
14 1488 4641 10478 1297 759 1028
16 1191 3554 7998 1044 565 763
18 976 2780 6448 847 498 637
20 238 2323 5152 699 403 516
Slope 1.786 1.942 1.979 1.875 1.841 1.900

Conclusion

The outlines of the leaves tested had fractal dimension ranging from 1.7 to 1.9. From this analysis, one can approximate that leaves are almost two dimensional objects.

Sources of Error

Some of the juniper leaves were overlapping during the scan and this would certainly affect the raw data by decreasing the amount of boxes needed to outline the leaf. Another source of error would the inclusion of the stems in the leaf images. A minor source of error is when the scanned image was analyzed using Fractal Dimensions, the program calculated the number of boxes needed to fill the surrounding space around the leaf rather than just the outline of the leaf itself.

Michael Ng, Richard Lau, Lisa Tsang, Ricky Wells -- 2002

Chaos Project