Another one of the pioneers of chaos theory, Benoit Mandelbrot, a mathematician at IBM, used yet another mathematical technique. In his capacity as a researcher for IBM, he looked for – and found –”patterns” in a wide variety of natural “random” processes. He found, for instance, that the background “noise” that is always present in telephone transmissions, follows a pattern that is completely unpredictable, or chaotic, but is nevertheless mathematically definable. Using a computer at IBM, Mandelbrot was able to produce chaotic systems graphically, yet only using the simplest mathematical rules. These pictures, known as “Mandelbrot sets”, showed an infinite complexity, and when a computer drawing was “blown up” to show finer detail, the vast, seemingly limitless variety continued.
The Mandelbrot sets have been described as possibly the most complex mathematical object or model ever seen. Yet within its structure, there were still patterns. By repeatedly “magnifying” the scale and looking at finer and finer detail (something the computer could do indefinitely because the whole structure was based on a given set of mathematical rules) it could be seen that there were regular repetitions – similarities – at different scales. “The degree of irregularity” was the same at different scales. Mandelbrot used the expression “fractal” to describe the patterns evident within the irregularity. He was able to construct a variety of fractal shapes, by slightly altering the mathematical rules. Thus he was able to produce a computer simulation of a coastline which, at any scale (at any magnification) always exhibited the same degree of “irregularity” or “crinkliness”.
Mandelbrot compared his computer-induced systems to examples of geometries that were also fractal shapes, repeating the same pattern over and over again on different scales. In the so-called Menger Sponge, for example, the surface area within it approaches infinity, while the actual volume of the solid approaches zero. Here, it is as if the degree of irregularity corresponds to the “efficiency” of the sponge in taking up space. That may not be as far fetched as it may sound because, as Mandelbrot showed, there are many examples of fractal geometry in nature. The branching of the windpipe to make two bronchiole and their repeated branching right down to the level of the tiny air passages in the lungs follows a pattern that can be shown to be fractal. In the same way it can be shown that the branching of blood vessels is fractal. In other words, there is a “self-similarity”, a repeating geometric pattern of branching, at whatever scale is examined.
The examples of fractal geometry in nature are almost limitless and in his book, The Fractal Geometry of Nature, Mandelbrot sought to demonstrate just that. It has been found that the spectrum of the timing of a normal heartbeat follows fractal laws, perhaps due to the fractal arrangement of nerve fibres in the heart muscle. The same is true of the rapid involuntary eye movements that are a feature of schizophrenia. Thus, fractal mathematics is now routinely used in a variety of scientific fields, including physiology and disciplines as widely separated as earthquake studies and metallurgy.
Yet another indication of the deterministic basis of chaos has been shown in studies of phase transitions and by the use of what mathematical modellers call “attractors”. There are many examples of phase transitions. It can mean the change from the smooth “laminar” flow of a fluid to turbulent flow, the transition from solid to liquid or liquid to gas, or the change within a system from conductivity to “superconductivity”. These phase transitions may have crucial consequences in technological design and construction. An aircraft, for example would lose lift if the laminar airflow over the wing became turbulent; likewise, the pressure needed to pump water will depend on whether or not the flow in the pipe is turbulent.
The use of phase-scale diagrams and attractors represents yet another mathematical device that has found a wide variety of applications in apparently random systems. As in the case of other chaos studies, there has been the discovery of common patterns, in this case “strange attractors” in a variety of research programmes, including electric oscillators, fluid dynamics and even in the distribution of stars in globular clusters. All these various mathematical devices – period doubling; fractal geometry; strange attractors – were developed at different times by different researchers to examine chaotic dynamics. But all their results point in the same direction: that there is an underlying mathematical lawfulness in what was always considered to be random.
A mathematician, Mitchell Feigenbaum, pulling a number of threads together, has developed what he has called a “universal theory” of chaos. As Gleick says “he believed that his theory expressed a natural law about systems at the point of transition between order and turbulence…his universality was not just qualitative, it was quantitative…it extended not just to patterns but to precise numbers.”
Marxists would recognise here the similarity with the dialectical law known as the law of transformation of quantity to quality. This idea describes the transition between one period of more or less gradual development, when change can be measured or “quantified”, and the next when change has been so “revolutionary”, there has been such a “leap”, that the entire “quality” of the system has been altered. Gleick’s use of the terms in a similar sense here is yet another indication of the way modern scientific theory is stumbling towards materialist dialectics.
The central point about the new science is that it deals with the world as it really is: as a constantly shifting dynamic system. Classical linear mathematics is like formal logic, which deals with fixed and unchanging categories. These are good enough as approximations, but do not reflect reality. Dialectics, however, is the logic of change, of processes and as such it represents an advance on formalism. In the same way, chaos mathematics is a step forward from the rather “unreal” science that ignored uncomfortable irregularities of life.