|A geometric form exhibiting the property of \'self-similarity\'; that is, a part of the form has the same properties as the form as a whole. If an image is fractal, then any part of the image if suitably enlarged will be indistinguishable from the image as a whole. Certain regular geometric forms have this property, but of greater interest in geography are forms that possess the property in a statistical sense, meaning that a part of the form has the same statistical properties as the whole. Parts of the coastline of Britain, for example, look similarly contorted at different scales, although the exact details are clearly different. If the same or similar forms exist at all levels, it follows that an observer cannot determine the scale at which the form is depicted.
The most important measure of a fractal form is its \'fractional dimension\'. A line is normally a one-dimensional object, and if magnified by a factor of 2 its length will double. But a fractal form reveals more detail when magnified, and its length thus grows disproportionately faster. Unlike normal lines, a fractal line has a fractional dimension greater than 1. In the limit an infinitely wiggly line fills the space that contains it and behaves as if it had the dimensions of the space; thus an infinitely wiggly line on a sheet of paper quadruples rather than doubles in length when both dimensions of the paper are doubled. Mandelbrot (1982) argued that geographical landscapes and many other aspects of natural systems exhibit fractal properties, and demonstrated that such phenomena exhibit regularities across a range of scales. He found, for example, a dimension of approximately 1.2 for the west coast of Britain. A large literature has evaluated Mandelbrot\'s assertion statistically, with mixed results (Xia and Clarke, 1997). Nevertheless, the fractal concept provides a useful framework for many investigations of geographical form. Batty and Longley (1994) have used fractal concepts as the basis for their models of urban form and growth, arguing that cities reveal detail at many scales, and that some aspects of the forms of cities are self-similar. Goodchild and Mark (1987) describe several implementations of fractal concepts within the technical design of geographic information systems, including quadtrees and spatial indexing schemes.Â (MG)
References Batty, M. and Longley, P.A. 1994: Fractal cities: a geometry of form and function. London: Academic Press.Â Goodchild, M.F., and Mark, D.M. 1987: The fractal nature of geographic phenomena. Annals of the Association of American Geographers 77: 265-78.Â Mandelbrot, B.B. 1982: The fractal geometry of nature. San Francisco: W.H. Freeman.Â Xia, Z.-G. and Clarke, K.C. 1997: Approaches to scaling of geo-spatial data. In D.A. Quattrochi and M.F. Goodchild, eds, Scale in remote sensing and GIS. Boca Raton: Lewis Publishers.