The fractal concept introduced into anthropology by Haraway and subsequent works by Wagner, McWhinney, and Strathern are epitomized by the idea of the sandy beach. We begin the description of our model by (1) recalling this static concept, then (2) we will extend it to the dynamical model of Lewin, Thom, and Zeeman as the fractal seraratrix or basin boundary, and finally (3) we will use the model to introduce the concept of the dischaotic personality.
In Mandelbrot's classic text, the second chapter is titled: 'How long is the coast of Britain?' We will describe the sandy beach in the two-dimensional con text of a map. Thus, the ocean and the land are mostly two- dimensional. Before fractal geometry, the map showed the boundary between the ocean and the land as a smooth curve: a one-dimensional coast. But now, thanks to Mandelbrot (he gives credit to Richardson), we may zoom in on the coast, and see that it has very small islands, even pebbles, in a densely packed structure. Zooming in again, we see grains of sand on the beach, and in the ocean close to the beach. All this is the coast: it has a fractal dimension. Land penetrates into the ocean in a frothy structure of sand, ocean penetrates into the land in a frothy structure of water in the wet sand. Not only is the coast a fractal, with a dimension more than one but less than two, but it is a fractal region: the coastal zone. The ocean and land are not divided by the coast in a binary fashion: they interpenetrate in a fractal geometry. The fractals of chaos theory (attractors, separatrices, and bifurcations) are all of the sandy beach variety.
We now make a jump to the dynamical model of Lewin, who imagined the life space or psychological field of a person as the state space of a (continuous) dynamical system. The observable behaviors in this model are the attractors, and the significant regions of life space, then, are the basins of attractions of these attractors. Further, the separatrices (the boundaries of these basins) are crucial to the Lewinian view of psychology [vanilla ]. In many important examples, these separatrices are fractal [ueda road to chaos ], [yorke wada ]. This means that the sandy beach concept applies to the boundary between two different behavioral regions. This will be our basic model in this paper. We should point out, however, that the improvement of the Lewinian model due to Thom and Zeeman is more complex: the attractor-basin portrait in the state space (life space of Lewin) is replaced by the response diagram, in the product of the state space and the control space of a dynamical scheme (morphogenetic field of Thom.)
We now assume a Lewinian dynamical model for the 'self' or life space of an individual. Different aspects of the personality, depending critically on the individual, are represented in this model by groups of basins of attraction. These may be slowly changing in time, under the effects of learning, adaptation, stimuli, and so on. Now that chaos theory and fractal geometry have emerged, we expect that fractal boundaries of these psychological regions are the rule, rather than the exception. Following the lead of chaos theoretic models in medical physiology, we may expect that chaotic attractors and fractal separatrices are important for health. Specifi cally, we may suggest that fat fractal separatrices in the psyche have an integrating effect. For under the effect of random or chaotic stimuli, the trajectory of the Lewinian model jumps about in small discontinuities, landing in different basins because of the fractal boundaries. This has the effect of integrating the different behaviors of the different attractors into a strongly associated or mixed personality. On the other hand, when the boundaries have become (perhaps in a pathological situation) too ordered (or dischaotic) or if the fractal dimension is too small, there would be a ten dency to manifest one attractor for some time, until an exceptional stimulus pushes the trajectory over the edge into the basin of another aspect of the self, and there is a dramatic change in behavior. Posing dischaos/chaos as a binary dichotomy instead of order/disorder, we may call this situation personality dischaos, rather than the more patriarchal personality disorder.