Chaos: Making a New Science

Benoit Mandelbrot began to question the standard scientific method of using the bell curve to track averages within systems, particularly economic systems. His research and intuition led him to believe that the geometric patterns of dynamic systems were far more complex. Economists habitually dismissed small fluctuations in prices or demand in favor of looking for larger patterns; the first were random, and thus insignificant, in comparison to long-term trends. However, as Mandelbrot and others showed, both short-term inconsistencies and long-term trends are crucial to understanding how a system works as a whole. Mandelbrot’s breakthrough, using IBM’s early computers, was discovering that measurements looked different at different scales. What appeared to be random when viewed at large scales actually showed a clear pattern when viewed at smaller scales. Additionally, Mandelbrot believed that this phenomenon did not just appear in economics but was a “signature” of nature, of natural systems (86).

Mandelbrot was something of an iconoclast throughout his career, defying specialization. After analyzing economic data, he tackled another problem of specific concern to communications (and thus to IBM, his employer): the noise that inevitably occurred during transmissions. Mandelbrot argued that tracking this supposedly random noise would reveal a consistent pattern: Within the bursts of noise, periods of noise-free transmission would always occur. This can be explained mathematically using an abstraction called the Cantor set, named for 19th-century scholar Georg Cantor. Essentially, the Cantor set exemplifies fractal time, showing that within every localized time frame (hours, minutes, seconds), the amount of noise to the amount of clean transmission remains constant. Thus, as in May’s experiments with population numbers, Mandelbrot’s theories suggest both discontinuity and constancy within dynamic systems.

Mandelbrot then turned his attention to geometry. Traditional Euclidean geometry represents shapes as ideal, consistent and mathematically stable. This kind of geometry, Mandelbrot claims, bears no resemblance to the geometry of nature, of natural systems, which is “rough, not rounded, scabrous, not smooth” (94). Mandelbrot extrapolates from this notion to propose the question about the length of coastlines: Any coastline, he argues, is infinite; the smaller the scale of measurement, the more twists and turns one can find in the irregular geometry of the border. The land mass it contains, however, is finite. Thus, Mandelbrot’s geometry is fractal, containing a finite area within an infinite boundary (See: Index of Terms). Another demonstration of this natural geometry is the Koch snowflake, wherein each arm of the snowflake can branch into an infinite number of smaller branches while remaining basically consistent in area. Fractals are “self-similar,” reproducing patterns within patterns in a “recursive” manner (103). Mandelbrot derived his intuitions from common-sense observations, as in the “infinitely deep reflection of a person standing between two mirrors” (103).

Christopher Scholz began using Mandelbrot’s ideas to describe geophysical properties, such as the earth’s surface (which ultimately helped in understanding earthquakes). When viewed from a great distance, the earth’s surface appears smooth and uniform; however, the closer one gets to that surface, the more one observes the nooks and crannies, the crags and bumps. That uniform surface begins to appear randomly chaotic. Using fractal geometry, Scholz observed that “surfaces in contact do not touch everywhere” (106). The irregularity of surfaces prevents that kind of uniformity; thus, fluid can always flow in between miniscule crevices. Like Humpty-Dumpty, a broken cup can never be put back together again; there will always be gaps, even if the cup looks whole from a distance. To Scholz, fractal geometry “gives you the mathematical and geometrical tools to describe and make predictions” (107). It functions better than traditional Euclidean geometry in understanding the dynamic processes that make up the earth’s surface.

Fractal geometry shows up everywhere in nature: in blood vessels, which endlessly branch, and in forests, where trees produce fractal leaves. These fractals appear overly complex only in comparison to Euclidean geometry. While other scientists sometimes derided Mandelbrot, it became clearer over time that his geometric revelations were better suited to understanding natural systems than Euclidean geometry. The importance of scaling became key to the science of chaos. This trend in science echoed larger trends in the culture during the 1960s and 1970s, as art and architecture left strictly Euclidean shapes behind and a yearning for a return to the wilderness gained strength in the face of modernity.

In contrast to Mandelbrot’s approach to understanding nature through fractal geometry, physicists wanted to understand the underlying forces; not what structures look like or how systems behave but why. Turning to the problem of turbulence allows for another way of understanding how complex systems work. Turbulence is problematic because it “is a mess of disorder at all scales” (122), not to mention that the border between stability and turbulence appears highly unpredictable. Theoretical physicists had arrived at this understanding but could get no further.

Physicist Harry Swinney, however, wanted to experiment with actual materials. He wanted to understand more about phase transitions, as when a liquid turns to gas or when a smooth system becomes turbulent. He constructed an experiment to show the flow of water across cylinders; previous theories had suggested that as the rate of flow increased, so would the frequency, in tandem. However, Swinney’s experiment revealed that, after a certain point, the increased rate of flow produced chaotic frequency; “no gradual buildup of complexity” (131) occurred. The theory did not match the reality.

David Ruelle, a Belgian physicist, was also working on this problem. He and a colleague had published a paper in which they proposed the existence of something they called a strange attractor (See: Index of Terms). Within phase space—wherein a dynamic system is captured as a single point in one moment of time—the strange attractor pulls the system toward it. Phase space allows for motion to be explored more fully, illuminating dynamic systems with more clarity. Turbulence suggests that dynamic systems contain “infinite modes, infinite degrees of freedom, infinite dimensions” (137), and thus infinite possibilities for chaos. Thus, such a process remains unmeasurable unless one uses fractals. The strange attractor becomes the stable point around which the infinitely dynamic orbit coalesces within a finite amount of space. This is demonstrated by the Lorenz attractor, which shows the infinite looping of an orbit that would never intersect: It looks somewhat like a set of butterfly wings, which is why Lorenz called it the butterfly effect.

In attempting to look more closely at strange attractors, scientists began to use Poincare maps, which take a section of the orbit and turn it into a set of points. This begins to reveal the order in the seeming disorder within the loops, their fractal structures. In looking at the solar system using computer models, scientists discovered that the orbits of the planets were irregular; in fact, sometimes these orbits became so unstable that it looked as if the planets might fly off into space. However, they always seemed to reorient back to some point. The irregular system had “clear remnants of order” (147). Strange attractors explain this. When examined more closely, what at first appear to be single lines in the orbits are actually pairs, and within these pairs are more pairs. Like nesting Russian dolls, an infinite sequence of lines exists within the orbital shape. This shape is that of a strange attractor, “the trajectory toward which all other trajectories converge” (150). Order exists within apparent disorder; the infinite is constrained within the finite. Randomness is not so random after all.