Fractal Analysis

Authored by: Mohammad Shafiur Rahman

Handbook of Food and Bioprocess Modeling Techniques

Print publication date:  December  2006
Online publication date:  December  2006

Print ISBN: 9780824726713
eBook ISBN: 9781420015072
Adobe ISBN:

10.1201/9781420015072.ch14

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Abstract

Fractal analysis is mainly applied when other methods fail or become tedious to solve complex or chaotic problems. Many natural patterns are either irregular or fragmented to such an extreme degree that Euclidian or classical geometry could not describe their form (Mandelbrot 1977, 1987). Any shape can be characterized by whether or not it has a characteristic length (Takayasu 1990). For example, a sphere has a characteristic length defined as the diameter. Shapes with characteristic lengths have an important common property of smoothness of surface. A shape having no characteristic length is called self-similar. Self-similarity is also known as scale-invariance, because selfsimilar shapes do not change their shape under a change of observational scale. This important symmetry gives a clue to understanding complicated shapes, which have no characteristic length, such as the Koch curve or clouds (Takayasu 1990). The idea of a fractal is based on the lack of characteristic length or on self-similarity. The word fractal is a new term introduced by Mandelbrot (1977) to represent shapes or phenomena having no characteristic length. The origin of this word is the Latin adjective fractus meaning broken. The English words “fractional” and “fracture” are derived from this Latin word fractus, which means the state of broken pieces being gathered together irregularly.

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