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Fractal Dimensions

Ecological Statistics

  1. Bai-Lian Li

Published Online: 15 SEP 2006

DOI: 10.1002/9780470057339.vaf015

Encyclopedia of Environmetrics

Encyclopedia of Environmetrics

How to Cite

Li, B.-L. 2006. Fractal Dimensions. Encyclopedia of Environmetrics. 3.

Author Information

  1. University of New Mexico, NM, USA

Publication History

  1. Published Online: 15 SEP 2006

Abstract

The term ‘fractal’ (from the Latin fractus, meaning ‘broken’), introduced by Benoit Mandelbrot about 25 years ago, is used to characterize spatial and/or temporal phenomena that are continuous but not differentiable. Geometrically, a fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractal structures do not have a single length scale, while fractal processes (e.g. time series) cannot be characterized by a single time scale. Fractal theory offers methods for describing the inherent irregularity of natural objects. In fractal analysis, the Euclidean concept of ‘length’ is viewed as a process. This process is characterized by a constant parameter D known as the fractal (or fractional) dimension. There are many fractal dimensions introduced in mathematical and physical literature. Fractal dimensions can be positive, negative, complex, fuzzy, multifractal, etc. The two most commonly used are the Hausdorf dimension and capacity.

Keywords:

  • shape;
  • self-similarity;
  • Hausdorf dimension;
  • MacArther-Wilson's model;
  • habitat