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Variation of Fractal Dimension of Leaves Based on Stem Position

Abstract

Utilization of methods based on Euclidean geometry to perform routine measurements of irregular objects could prove to be exceptionally difficult and particularly inefficient. These irregular arrangements such as leaf shapes are called fractals and are more efficiently described within the geometry of fractals.

The purpose of the experiment, in the present study, is to examine shapes of plant leaves in relation to their position on the stem in terms of fractal dimensions. The hypothesis suggests that fractal dimension does vary among the leaves located at various positions on the stem.

In this experimental study, five samples of Norfolk Island Pine Araucaria Heterophylla plants were obtained and were carefully deprived of their leaves. The fractal dimension of each leaf was determined using the box-count method. Five trials were conducted using five plants. The mean fractal dimensions of each leaf was obtained and then analyzed by ANalysis Of VAriance between groups [ANOVA].

I. Introduction

Shapes have always been an important aspect in biological systems. Although usually ignored, shapes play a major role in description of functions of various organisms. Traditionally, the shapes of objects and organisms have been described using Euclidean geometry1. Euclidean geometry describes the basic, regular figures that are most familiar such as lines, squares, cubes, etc. Irrespective of the case, all these structures have dimensions that are positive integers (whole numbers): 0 for a point, 1 for a line, 2 for a surface and 3 for volume2. However, objects do not always display these simple shapes, especially in nature.

The study of fractal dimension is currently being applied to almost every branch of science, mathematics3 and economics4. Its applications to medical and biological sciences have been extensive. Recent studies have shown that fractal geometry can be useful for describing the pathological architectures of tumors and, perhaps more surprisingly, for yielding insights into the mechanisms of tumor growth, i.e., angiogenesis, that complement those obtained by modern molecular methods5. In other cases, the study of fractal dimensions in dynamic systems such as the fluctuations of a human heart beat could lead to the detection of heart diseases depending on the irregularity of the heart beat frequency6. These fractal dimensions could be measured using the box count method.

Fractals are used in many applications across the sciences. One of the biggest advances in fractal application has been in the fields of image analysis and pattern recognition. From understanding facial expressions to modeling a pattern, fractals have made significant advancements.

In a paper by Iftekharuddin, et. al.7, fractals have been applied to brain tumors. Using three methods identified as piecewise modified box-counting, piecewise triangular prism surface area, and piecewise threshold box counting, images of the brain have been analyzed. The difference

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Misterio, J., & Ravindra, K., & Rivero, R. D., & McCloud, H., & Burr-Alexander, L., & Ravindra, N. (2007, June), Variation Of Fractal Dimension Of Leaves Based On Stem Position Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. 10.18260/1-2--2852