A Love Letter To Chaos Theory and Its Prophets

The Story of Chaos Theory and Some Fun Facts About the Scientists

This is part of a series of classic science books that I think should be widely read to help us have a better understanding of the world we live in, and of each other.

I read the book Chaos: Making a New Science, by James Gleick every couple of years. I even keep a first edition protected in its wrapping, it is such an important book that helped shape my thinking of the world. The dynamic writing of Gleick invites us into the complex alleys of chaos theory, and believe me there are many, it is an area of study that has upended traditional paradigms of predictability and order. This is a book that is not merely a history of a scientific concept but a tribute to those wise people who dared to defy the certainty of their times. Published in 1987, Gleick's narrative is as much about the science as it is about the enigmatic individuals whose curiosity led them to the heart of the unpredictable. Gleick presents a portrait of chaos as an elegant rebellion against the Newtonian worldview, revealing the profound order hidden within apparent randomness.

The Revolutionaries of Randomness

The story of chaos theory is not confined to laboratories or equations, it is the story of its protagonists. Gleick evokes the intellectual daring of figures like Edward Lorenz, a meteorologist who stumbled upon chaos while trying to predict the weather, and Benoit Mandelbrot, the father of fractals, whose audacity turned jagged coastline measurements into an entirely new way of viewing geometry. These scientists were iconoclasts, undeterred by the scorn of colleagues who clung to the linear, the predictable, and the comfortable. Lorenz's discovery, a seemingly innocuous realization that tiny changes in initial conditions could lead to vastly different outcomes, echoed through the scientific world like a thunderclap, forever altering our concept of causality. 

One notable example that Gleick elaborates on is Lorenz's accidental discovery: Lorenz was working on a rudimentary computer, the Royal McBee doing experiments with a simple weather model. By rounding a number from 0.506127 to 0.506, Lorenz discovered that his weather simulation yielded entirely different results, a revelation that would become the butterfly effect. This anecdote is a powerful illustration of chaos theory's foundational principle: that deterministic systems can still harbor unpredictable behaviors.

And such is life…

Gleick moves seamlessly through the history of ideas, linking the deterministic past with an uncertain future. He takes us from Henri Poincaré, who first glimpsed the edges of chaos in his celestial mechanics, to Mitchell Feigenbaum, whose numerical precision uncovered universal constants hidden within turbulence. These pioneers were united by a sense of wonder at the unpredictability of the natural world. They saw beauty in the irregular, a beauty that eluded the grasp of traditional mathematics. Fractals, Mandelbrot's jagged children, are portrayed not as abstract curiosities but as reflections of nature itself, from the branching of trees to the rhythm of the human heart.

Gleick provides an example involving heartbeat rhythms, explaining how slight fluctuations in a healthy heart rate are an indication of resilience, whereas too much regularity could signal underlying issues. This understanding offers a practical perspective of chaos, one that connects abstract theory to tangible outcomes in human health. Gleick shares a vivid anecdote about Mandelbrot's fascination with natural forms, recounting how Mandelbrot measured the jagged coastline of Britain and arrived at a different result each time. This led to his pioneering work in fractal geometry, a geometry of roughness that revealed an underlying order in irregular structures. The fractal dimension, a new kind of metric, was Mandelbrot's answer to describing forms that defied classical Euclidean geometry.

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

~ Benoit Mandelbrot

Hidden Figures: The Women of Chaos Theory

Research has revealed that women like Ellen Fetter and Margaret Hamilton made significant contributions to the early development of chaos theory through their work in programming the computers used for crucial simulations. Ellen Fetter, for instance, was instrumental in the computational work behind Lorenz's discovery of the butterfly effect, managing the complex calculations that brought chaos to life. Margaret Hamilton, later famous for her work on NASA's Apollo missions, also played a key role in developing the rigorous programming techniques that made it possible to explore nonlinear systems effectively.

These contributions were largely unsung for decades, overshadowed by the more prominent figures of Lorenz, Mandelbrot, and others. Only relatively recently have Fetter and Hamilton received the recognition they deserve for their pioneering work. Their efforts were crucial in bringing about the computational era that enabled chaos theory to flourish, proving that the story of chaos was not just one of bold ideas, but also of meticulous, often painstaking, programming.

Beyond those directly involved in the research Gleick wrote about, he sadly lessens and even ignores there roles. Other mathematicians and scientists whose work laid the groundwork for chaos theory, such as Sonya Kovalevskaya and Mary Cartwright, should also be rightly recognized.

The Missing Pieces of Chaos

Yet, for all its grandeur, chaos theory is a puzzle still lacking crucial pieces. Gleick acknowledges that while chaos provides a framework for understanding complex systems, such as weather patterns, the beating of the heart, or financial markets, the translation of these abstract insights into practical tools remains, well let’s say challenging! The difficulty of predicting ecological dynamics, or the intricate workings of the human brain, underscores a gap that science has yet to reveal. Nevertheless, the promise of chaos theory remains tantalizing, and while significant advances have been made, fully harnessing its potential still requires further synthesis. With the advent of machine learning, deep neural networks, and computational power, scientists have developed more sophisticated ways to model chaotic systems, making progress in fields like meteorology, cardiology, and even climate science. However, the quest to create a comprehensive mathematical framework that can make chaos predictably useful across a diverse range of disciplines is ongoing. Researchers continue to explore how best to move from theoretical chaos to practical, actionable insights.

My one quibble with Gleick's work, while masterful in scope and storytelling, he sometimes glosses over the limitations of chaos theory's application. There is an implicit bias towards presenting chaos as universally applicable, while the truth is that some systems resist quantification even with the advanced tools chaos theory provides. I believe that more nuanced discourse would enrich the reader's understanding. 

From Weather to Wall Street

As I've noted, Chaos theory's influence spans many disciplines, from meteorology to finance. Gleick captures the exuberance of the late 20th-century scientific community as it discovered fractals in the cotton prices of Wall Street and in the arteries of human physiology. Mitchell Feigenbaum's bifurcation diagrams are presented not merely as abstract constructs but as the skeleton key to understanding how order and disorder coexist within deterministic systems. Yet Gleick is careful to temper this enthusiasm with realism. The scientists of chaos were not prophets predicting an inevitable future; they were explorers on a path that stubbornly refused to be tamed.

One compelling anecdote involves financial markets, where traders saw echoes of chaos in price movements that resisted classical statistical analysis. By applying fractal models, analysts began to see the unpredictability of markets in a new light, though the application was still fraught with complications and fell short of providing true predictability. I recommend reading Mandelbrot's The (Mis)Behavior of Markets for more insights on this fascinating area.

A Universe Writ Small

Gleick draws us to the realization that chaos is not an esoteric domain but an essential feature of our everyday existence. The book takes us from the grand, the swirling dynamics of distant galaxies, to the intimate: the patterns of heartbeat irregularities. Through Gleick’s deft storytelling, I sense that chaos is not merely a theory; it is a way of seeing the world that recognizes the fragile balance between stability and change. Chaos suggests that systems, be they physical, biological, or social, are woven with threads of both determinism and unpredictability. The clockwork universe envisioned by Newton, with all its gears and cogs, is replaced by something infinitely richer, a universe where the flutter of a butterfly's wings in Brazil can set off a tornado in Texas.

Interestingly, Lorenz wasn't trying to create a catchy phrase; the idea emerged from a scientific talk in which he aimed to convey the sensitivity of chaotic systems to initial conditions. It was Gleick's vivid storytelling that helped make this abstract scientific concept relatable and iconic to the general public, eventually being referenced in movies, literature, and general conversations.

Chaos and Complexity

Mitchell Feigenbaum, one of the key figures in chaos theory, famously used a simple, inexpensive handheld calculator to derive his groundbreaking constants related to period-doubling bifurcations. Working late nights at Los Alamos National Laboratory, Feigenbaum painstakingly computed numbers, discovering a universal constant that would later bear his name. His determination to solve complex problems with the simplest tools available serves as a reminder that great discoveries often emerge not from cutting-edge technology but from curiosity, creativity, and persistence.

That being said, chaos theory still lacks the computational precision needed to make sense of the biological or neurological realms in a predictive manner. Our tools, sophisticated as they are, still fall short when tasked with reducing the unfathomable intricacies of the human mind or global ecosystems into something manageable. The challenge that chaos theory presents is not just one of understanding, but one of synthesis, bringing together physics, biology, psychology, and technology in a way that discovers rather than obscures.

This is perhaps where Gleick could have gone deeper, providing a more critical analysis of the limits of chaos. By emphasizing the continued struggle to apply chaos theory practically, whether in medicine, ecology, or economics, he might have offered readers a more balanced view of both its potential and its constraints. Such a critical lens would enhance the reader's appreciation of the complexities inherent in this "new science."

Quirky Chaos

Some fun and quirky anecdotes related to chaos theory and the scientists featured in James Gleick's Chaos:

1. The Butterfly That Was Almost a Seagull: The term "butterfly effect" is iconic now, but it almost wasn’t a butterfly at all. In his early drafts, Edward Lorenz considered other metaphors to describe how a tiny change could have far-reaching effects. At one point, he even used a seagull to illustrate the idea. Imagine if today we talked about “the seagull effect”! The butterfly eventually stuck, perhaps because it was more poetic and evocative, and it turned out to be a perfect metaphor for chaos theory's sensitivity to initial conditions.

2. Feigenbaum's Obsession with Junk Food: Mitchell Feigenbaum, one of the leading figures in chaos theory, had a bit of a reputation for his quirky eating habits. He often worked late at night, into the early hours of the morning at Los Alamos, fueled largely by junk food and coffee. It’s said that during his intense periods of research, his diet consisted primarily of chocolates and french fries, something he didn't see as a problem, given his laser focus on discovering universal constants.

3. Mandelbrot’s Early Visual Displays: Benoit Mandelbrot used to carry around hand-drawn visuals of fractals, which he would pull out during seminars to explain his ideas. In the early days, computers didn’t have the capability to create sophisticated graphics, so Mandelbrot had to illustrate the complexity of fractals by hand. Later, he managed to use IBM’s advanced (for its time) computing resources to generate fractal images. When those beautiful fractals were finally displayed on computer screens, they captivated scientists and laypeople alike, sparking immense excitement and even admiration in art circles.

4. Weather Prediction through Toy Demonstrations: Edward Lorenz famously loved using analogies and simple tools to convey complex ideas. One of his favorite teaching tools was apparently a small toy, a kind of physical metaphor for the unpredictability of weather patterns. He would often demonstrate how chaotic patterns emerged in even the simplest water currents, which helped audiences grasp why weather prediction was so inherently difficult. Lorenz had a knack for finding practical, often playful ways to illustrate sophisticated principles, helping people “see” chaos firsthand.

5. A "Fractal Face-Off" Between Mandelbrot and Euclid: Mandelbrot’s fractals turned the orderly world of Euclidean geometry on its head, but the difference between the two also led to some funny exchanges during his lectures. Mandelbrot used to joke that Euclidean geometry was for “flat” and boring things, like perfect circles and squares, whereas his fractal geometry was for "real" things—like broccoli and mountains. It’s said that during one of his lectures, he drew a simple line and said, “This is Euclid,” then drew a jagged, endlessly branching figure and declared, “This is me.” The audience loved it, and it made the stark difference between their worldviews abundantly clear.

6. Lorenz’s Tangle with his Royal McBee: The computer Lorenz used for his groundbreaking simulations, the Royal McBee, was notoriously temperamental. Lorenz often joked that getting useful results from it required more patience than a monk! It was basically an early, primitive computer that was prone to overheating and crashing. Lorenz’s patience with it is well-documented. It’s amusing to think that one of the foundational discoveries of chaos theory might never have happened if Lorenz hadn’t managed to coax the Royal McBee into cooperating just a little longer.

These fun anecdotes illustrate not just the brilliance of the scientists but also their human quirks, eccentricities, and the unconventional circumstances in which many of their breakthroughs occurred.

The Elegant Rebellion

Chaos: Making a New Science by James Gleick is more than a chronicle of a scientific revolution, it is a love letter to those who see beyond the linear and find enchantment in the unpredictable. Gleick's narrative pulsates with the energy of the scientists who broke rank, who found joy in exploring what others saw as formless confusion. Chaos theory, as portrayed by Gleick, is not a rejection of order but a deeper appreciation for the nuanced patterns that emerge from disorder. The book leaves us with a sense of wonder, a recognition that while we may not fully control the complexities of nature, we can at least begin to understand them.

For the reader, keen to understand both the science and the story of those who shaped it, Gleick provides an evocative narrative that shows us that science, at its best, is a collective exploration into the unknown.

Stay curious

Dr Colin W.P. Lewis

This is part of a series of classic science books that I think should be widely read to help us have a better understanding of the world we live in, and of each other.