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 |
Analysis
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| Maple -- Dimension: 1.98 | "Maple, Jr." -- Dimension: 1.94 |
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| Juniper -- Dimension: 1.79 | "Leaf 4" -- Dimension: 1.88 |
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| "Leaf 5" -- Dimension: 1.84 | "Leaf 8" -- Dimension: 1.90 |
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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