Chaos: Making a New Science

Another scientist interested in Goethe’s ideas, Albert Libchaber, wanted to create a physical experiment that “would reveal the onset of turbulence” (192). He called it “Helium in a Small Box,” wherein he heated liquid helium in order to trigger turbulence. Libchaber wanted better to understand flow, which can be defined as “shape plus change, motion plus form” (195). Thus far, the standard differential equations could not capture the “recursive power of flows within flows” (195). A link might exist between universality and motion, according to Libchaber and others at the time.

Libchaber also looked to the science of Theodor Schwenk, who was likewise interested in universality. Schwenk studied the flow of rivers in the early 20th century and speculated about what he called “sensitive chaos” in the currents; he recognized the currents within currents—that is, that flow left evidence of its movements behind. He likened these possibilities to how blood flowed within human organs—a universal standard within nature. D’Arcy Wentworth Thompson studied biology in the early 20th century, coming to similar conclusions: Patterns in nature are repeated in intuitive if unlikely ways. For example, a falling drop of ink through water looks remarkably similar to the structure of a jellyfish. Instead of studying these shapes and motions in isolation, Thompson sought a holistic explanation for these similarities. Thompson’s view argues that a Darwinian understanding of evolution opts mostly for a teleological vision—the organism evolves because its design is successful—rather than an underlying physical cause—the organism evolves because something in nature directs it in that manner. Thompson wanted to investigate the possibilities of both forces in the development of nature and life.

With these ideas in the background, Libchaber began his experiments with helium and flow. To chart the changes, Libchaber examined how the flow changed over time rather than in space. His experiments proved Feigenbaum’s theory: Predicting the patterns of frequencies, both in number and in strength, within the seeming randomness of flow, was possible. This experiment, in turn, led to a new era of discovery using computer modeling, which could help in the search for strange attractors. Nevertheless, real-world experiments like Libchaber’s remain important. While computers reflect the wishes of their programmers, nature often disrupts expectations.

Mathematician Michael Barnsley entered the field of chaos by suggesting that, underlying Feigenbaum’s theory is another “fractal object hidden from view” (215). By looking at a complex plane, comprising both longitudinal numbers (or “real numbers”) and latitudinal numbers (or “imaginary numbers”), one can look at results in two dimensions rather than one. When Barnsley tested these suppositions, he found a “fantastical family of shapes” that increase understanding of both dynamic systems and abstract mathematics (216). Another mathematician, John Hubbard, also started investigating some of these ideas. In particular, he wanted to prove the existence of Lorenz’s strange attractor via mathematical evidence. By using a computer to model the results of equations using imaginary numbers (which, despite the name, are just as “real” as real numbers), he discovered “boundaries of infinite complexity” (220). That is, he found that simple boundaries between two points do not exist on a complex plane; rather, a third always inserts itself, exposing another fractal shape.

This process eventually led to the Mandelbrot set, another paradox uncovered by chaos science (See: Index of Terms). While the Mandelbrot set contains an infinite number of images, it can also be transmitted across computer networks with a decidedly finite amount of code. The already complex shapes take on an even more complex character when scaled down, continuing to sprout new tendrils and offshoots. However, even within this infinite complexity, Mandelbrot could also recognize Feigenbaum’s theory at work; an observable pattern emerged within the sequencing. Unlike shapes in traditional geometry, the Mandelbrot set “allows no shortcuts” (226), no smoothing of rough edges, no rounding of numbers. This union between dynamic shapes and abstract mathematics yielded a new way of understanding geometry—a revolutionary development that would have been impossible without the computer, given its ability to refine points on a grid in various ways. Mandelbrot’s fractals were even more complex than he first thought, containing multitudinous variety within each subsequent piece of each fractal, ad infinitum.

Additionally, the Mandelbrot set revealed that boundaries are problematic, as the set is pulled between zero and infinity. In order to apply the Mandelbrot set to real-world interactions, boundaries must be identified, so that “the way a system chooses between competing options” (233) is predictable; this was the study of fractal basin boundaries. Scientists such as James Yorke found that predicting behavior near a boundary is impossible, even within nonchaotic systems.

Barnsley examined this problem through something he called “the chaos game” (236). One had to limit the set in order to determine the shape of the final system. Using an analogy of drawing a map of Great Britain, Barnsley noted that to incorporate every fractal twist and turn of the landscape would result in a final map so large that it would defeat the original purpose. These complexities must be examined at smaller scales using simple rules. In doing so via a computer, Barnsley demonstrated that fractals have both order and limitations encoded within them. That is, nature imposes restrictions upon itself; how a fern develops, for example, has a limited number of permutations. Even though Barnsley called his experiments a “game,” this did not imply randomness; the purpose of his game was to reveal the Mandelbrot set residing within the natural object. In this sense, the Mandelbrot set exists, a priori, before the thing itself.