Fractal

Fractals, a term coined by their originator Benoit Mandelbrot, are objects of any kind whose spatial form is nowhere smooth (i.e., they are irregular) and whose irregularity repeats itself geometrically across many scales. The irregularity of form is similar from scale to scale, and the objects are said to possess the property of self-similarity or scale invariance. A classic fractal structure is the Koch Island or snowflake (Figure 1). It can be described as follows:

• (a)
  Draw an equilateral triangle (an initial shape or initiator [Figure 1A]).
• (b)
  Divide each line that makes up the figure into three parts and “glue” a smaller equilateral triangle (a generator) onto the middle of each of the three parts (Figure 1B).
• (c)
  Repeat procedure (b) on each of the 12 resulting parts (4 per side of the original triangle).
• (d)
  Repeat procedure (b) on each of the 48 resulting parts (16 per side of the original triangle) and so on.

Figure 1 The Koch Island or Snowflake

This can ultimately result in an infinitely complex shape.

We use the term fractal dimension to measure fractals. The Euclidean dimensions are composed of 0 (points), 1 (straight lines), 2 (areas), and 3 (volumes). Fractal dimensions lie between these dimensions. Thus, a wiggly coastline (perhaps like each side of the Koch Island in Figure 1) fills more space than a straight line (Dimension 1) but is not so wiggly as to fill an area (Dimension 2). Its fractal dimension thus lies between 1 and 2. (The fractal dimension of each side of Figure 1 is actually approximately 1.262; the dimension of a more intricate, fjordlike coastline would be higher, closer to 2.) The tower blocks on the skyline of a city fill part of, but not all of, the vertical dimension, and so we can think of cities as having dimensions between 2 and 3.

The Koch Island shown in Figure 1 is a pure fractal shape because the shapes that are glued onto the island at each level of recursion are exact replicas of the initiator. The kinds of features and shapes that characterize our rather messier real world rarely exhibit perfect regularity, yet self-similarity over successive levels of recursion often can be established statistically. Just because recursion is not observed to be perfectly regular does not mean that the ideas of self-similarity are irrelevant. Christaller's central place theory provides one good example of a theory of[Page 172] idealized landscapes of nested hexagons that is applicable even though it is never observed exactly in realworld retail and settlement hierarchies.

Fractal ideas are important, and measures of fractal dimension can be as useful as spatial autocorrelation statistics or of medians and modes. Paul Longley and his colleagues described how the fractal dimension of an object may be ascertained by identifying the scaling relation between its length or extent and the yardstick that is used to measure it. Regression analysis provides one (of many) means of establishing this relationship. If we can demonstrate that an object is fractal, this can help us to identify the processes that give rise to different forms.

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