of this sum, and then we specify only the fractal statistics of the smaller lumps, thus fixing the qualitative appearance of the surface. Figure 4.8 illustrates an example of such description. The overall shape is that of a sphere; to this specified large-scale shape, smaller lumps were added randomly. The smaller lumps were added with three different choices of r (i.e., three different choices of fractal statistics) resulting in three qualitatively different surfaces--each with the same basic spherical shape.
The ability to fix particular "lumps" within a given shape provides an elegant way to pass from a qualitative model of a surface to a quantitative one--or vice versa. We can refine a general model of the class "a mountain" to produce a model of a particular mountain by fixing the position and size of the largest lumps used to build the surface, while still leaving smaller details only statistically specified. Or we can take a very specific model of a shape, discard the smaller constituent lumps after calculating their statistics, and obtain a model that is less detailed than the original but which is still qualitatively correct.
To support our reasoning abilities perception must recover environmental regularities for later use in cognitive processes. Understanding this recovery of structure is critically important because these regularities are the building blocks of all cognitive activities. To create a theory of how perception produces meaningful cognitive building blocks we need a representation whose elements may be lawfully related to important physical regularities, and that correctly describes the perceptual organization people impose on the stimulus. Unfortunately, the representations that are currently available were originally developed for other purposes (for example, physics, engineering) and have so far proven unsuitable for the problems of perception or commonsense reasoning.