Fractals

Fractals, a term coined by their originator Benoît Mandelbrot, in 1983, 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 object is said to possess the property of self-similarity; such objects are scale invariant. Many of the methods and techniques of geographic information science assume that spatial variation is smooth and continuous, except perhaps for the abrupt truncations and discrete shifts encountered at boundaries. Yet this is contrary to our experience, which is that much geographic variation in the real world is jagged and apparently irregular. Fractals provide us with one method for formally examining this apparent irregularity.

A classic fractal structure that exhibits the properties of self-similarity and scale invariance is the Koch Island or Snowflake (see Figure 1). It is described as follows:

• 1.
  Draw an equilateral triangle (an initial shape, or initiator: Figure 1A).
• 2.
  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).
• 3.
  Repeat Procedure #2 on each of the 12 resulting parts (4 per side of the original triangle: Figure 1C).
• 4.
  Repeat Procedure #2 on each of the 48 resulting parts (16 per side of the original triangle: Figure 1D); and so on.

This can ultimately result in an infinitely complex shape.

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 only rarely exhibit perfect regularity, yet self-similarity over successive levels of recursion can nevertheless often be established statistically. Just because recursion is not observed to be perfectly regular does not mean that the ideas of self-similarity are irrelevant: For example, Christaller's central place theory has led generations of human geographers to think of the hinterlands of small and larger settlements in terms of an idealized landscape of nested hexagonal market areas, although this organizing construct rarely, if ever, characterizes real-world retail or settlement hierarchies. The readings below provide illustrations of a range of other idealized fractal shapes and their transformation into structures that resemble elements of the real world.

Figure 1 The Koch Island or Snowflake