For decades, a small group of mathematicians has patiently unraveled the mystery of what was once math’s most popular picture. Their story shows how technology transforms even the most abstract mathematical landscapes.
Introduction
In the mid-1980s, like Walkman cassette players and tie-dyed shirts, the buglike silhouette of the Mandelbrot set was everywhere.
Students plastered it to dorm room walls around the world. Mathematicians received hundreds of letters, eager requests for printouts of the set. (In response, some of them produced catalogs, complete with price lists; others compiled its most striking features into books.) More tech-savvy fans could turn to the August 1985 issue of Scientific American. On its cover, the Mandelbrot set unfolded in fiery tendrils, its border aflame; inside were careful programming instructions, detailing how readers might generate the iconic image for themselves.
By then, those tendrils had also extended their reach far beyond mathematics, into seemingly unrelated corners of everyday life. Within the next few years, the Mandelbrot set would inspire David Hockney’s newest paintings and several musicians’ newest compositions — fuguelike pieces in the style of Bach. It would appear in the pages of John Updike’s fiction, and guide how the literary critic Hugh Kenner analyzed the poetry of Ezra Pound. It would become the subject of psychedelic hallucinations, and of a popular documentary narrated by the sci-fi great Arthur C. Clarke.
The Mandelbrot set is a special shape, with a fractal outline. Use a computer to zoom in on the set’s jagged boundary, and you’ll encounter valleys of seahorses and parades of elephants, spiral galaxies and neuron-like filaments. No matter how deep you explore, you’ll always see near-copies of the original set — an infinite, dizzying cascade of self-similarity.
That self-similarity was a core element of James Gleick’s bestselling book Chaos, which cemented the Mandelbrot set’s place in popular culture. “It held a universe of ideas,” Gleick wrote. “A modern philosophy of art, a justification of the new role of experimentation in mathematics, a way of bringing complex systems before a large public.”
The Mandelbrot set had become a symbol. It represented the need for a new mathematical language, a better way to describe the fractal nature of the world around us. It illustrated how profound intricacy can emerge from the simplest of rules — much like life itself. (“It is therefore a real message of hope,” John Hubbard, one of the first mathematicians to study the set, said in a 1989 video, “that possibly biology can really be understood in the same way that these pictures can be understood.”) In the Mandelbrot set, order and chaos lived in harmony; determinism and free will could be reconciled. One mathematician recalled stumbling across the set as a teenager and seeing it as a metaphor for the complicated boundary between truth and falsehood.
The Mandelbrot set is an infinitely intricate fractal shape that continues to captivate mathematicians decades after its discovery.
Maths.Town
Introduction
The Mandelbrot set was everywhere, until it wasn’t.
Within a decade, it seemed to disappear. Mathematicians moved on to other subjects, and the public moved on to other symbols. Today, just 40 years after its discovery, the fractal has become a cliché, borderline kitsch.
But a handful of mathematicians have refused to let it go. They’ve devoted their lives to uncovering the secrets of the Mandelbrot set. Now, they think they’re finally on the verge of truly understanding it.
Their story is one of exploration, of experimentation — and of how technology shapes the very way we think, and the questions we ask about the world.
The Bounty Hunters
In October 2023, 20 mathematicians from around the world congregated in a squat brick building on what was once a Danish military research base. The base, built in the late 1800s in the middle of the woods, was tucked away on a fjord on the northwest coast of Denmark’s most populous island. An old torpedo guarded the entrance. Black-and-white photos, depicting navy officers in uniform, boats lined up at a dock, and submarine tests in progress, adorned the walls. For three days, as a fierce wind whipped the water outside the windows into frothing whitecaps, the group sat through a series of talks, most of them by two mathematicians from Stony Brook University in New York: Misha Lyubich and Dima Dudko.
In the workshop’s audience were some of the Mandelbrot set’s most intrepid explorers. Near the front sat Mitsuhiro Shishikura of Kyoto University, who in the 1990s proved that the set’s boundary is as complicated as it can possibly be. A few seats over was Hiroyuki Inou, who alongside Shishikura developed important techniques for studying a particularly high-profile region of the Mandelbrot set. In the last row was Wolf Jung, the creator of Mandel, mathematicians’ go-to software for interactively investigating the Mandelbrot set. Also present were Arnaud Chéritat of the University of Toulouse, Carsten Petersen of Roskilde University (who organized the workshop), and several others who had made major contributions to mathematicians’ understanding of the Mandelbrot set.
The mathematicians Misha Lyubich (right) and Dima Dudko have spent decades exploring the Mandelbrot set.
Karen Dias for Quanta Magazine
Introduction
And at the whiteboard stood Lyubich, the world’s foremost expert on the topic, and Dudko, one of his closest collaborators. Together with the mathematicians Jeremy Kahn and Alex Kapiamba, they have been working to prove a long-standing conjecture about the geometric structure of the Mandelbrot set. That conjecture, known as MLC, is the final obstacle in the decades-long quest to characterize the fractal, to tame its tangled wilderness.
By building and sharpening a powerful set of tools, mathematicians have wrestled control of the geometry of “almost everything in the Mandelbrot set,” said Caroline Davis of Indiana University — except for a few remaining cases. “Misha and Dima and Jeremy and Alex are like bounty hunters, trying to track down these last ones.”
Lyubich and Dudko were in Denmark to update other mathematicians on recent progress toward proving MLC, and the techniques they’d developed to do so. For the past 20 years, researchers have gathered here for workshops dedicated to unpacking results and methods in the field of complex analysis, the mathematical study of the kinds of numbers and functions used to generate the Mandelbrot set.
It was an unusual setup: The mathematicians ate all their meals together, and talked and laughed over beers into the wee hours. When they finally did decide to go to sleep, they retired to bunk beds or cots in small rooms they shared on the second floor of the facility. (Upon our arrival, we were told to grab sheets and pillowcases from a pile and take them upstairs to make our beds.) In some years, conference-goers brave a swim in the frigid water; more often, they wander through the woods. But for the most part, there’s nothing to do except math.
Typically, one of the attendees told me, the workshop attracts a lot of younger mathematicians. But that wasn’t the case this time around — perhaps because it was the middle of the semester, or, he speculated, because of how difficult the subject matter was. He confessed that at that moment, he felt a bit intimidated about the prospect of giving a talk in front of so many of the field’s greats.
Alex Kapiamba (left) and Jeremy Kahn often work together on a blackboard in Kahn’s backyard, near Brown University’s campus in Providence, Rhode Island, as they try to gain control of the geometry of the Mandelbrot set.
Adam Wasilewski for Quanta Magazine
Introduction
But given that most mathematicians in the broader area of complex analysis are no longer working on the Mandelbrot set directly, why dedicate an entire workshop to MLC?
The Mandelbrot set is more than a fractal, and not just in a metaphorical sense. It serves as a sort of master catalog of dynamical systems — of all the different ways a point might move through space according to a simple rule. To understand this master catalog, one must traverse many different mathematical landscapes. The Mandelbrot set is deeply related not just to dynamics, but also to number theory, topology, algebraic geometry, group theory and even physics. “It interacts with the rest of math in a beautiful way,” said Sabyasachi Mukherjee of the Tata Institute of Fundamental Research in India.
To make progress on MLC, mathematicians have had to develop a sophisticated set of techniques — what Chéritat calls “a powerful philosophy.” These tools have garnered much attention. Today, they constitute a central pillar in the study of dynamical systems more broadly. They’ve turned out to be crucial for solving a host of other problems — problems that have nothing to do with the Mandelbrot set. And they’ve transformed MLC from a niche question into one of the field’s deepest and most important open conjectures.
Lyubich, the mathematician arguably most responsible for molding this “philosophy” into its current form, stands tall and straight, and speaks quietly. When other mathematicians at the workshop approach him to discuss a concept or ask a question, he closes his eyes and listens attentively, his thick eyebrows furrowed. He answers carefully, in a Russian accent.
Lyubich has nurtured generations of mathematicians at the institute he now runs at Stony Brook University.
Karen Dias for Quanta Magazine
Introduction
But he’s also quick to break into loud, warm laughter, and to make wry jokes. He’s generous with his time and advice. He has “really nurtured quite a few generations of mathematicians,” said Mukherjee, one of Lyubich’s former postdocs and a frequent collaborator. As he tells it, anyone interested in the study of complex dynamics spends some time at Stony Brook learning from Lyubich. “Misha has this vision of how we should go about a certain project, or what to look at next,” Mukherjee said. “He has this grand picture in his mind. And he is happy to share that with people.”
For the first time, Lyubich feels he’s able to see that grand picture in its totality.
The Prize Fighters
The Mandelbrot set began with a prize.
In 1915, motivated by recent progress in the study of functions, the French Academy of Sciences announced a competition: In three years’ time, it would offer a 3,000-franc grand prize for work on the process of iteration — the very process that would later generate the Mandelbrot set.
Iteration is the repeated application of a rule. Plug a number into a function, then use the output as your next input. Keep doing that, and observe what happens over time. As you continue to iterate your function, the numbers you get might rapidly rise toward infinity. Or they might be pulled toward one number in particular, like iron filings moving toward a magnet. Or end up bouncing between the same two numbers, or three, or a thousand, in a stable orbit from which they can never escape. Or hop from one number to another without rhyme or reason, following a chaotic, unpredictable path.
In the 1910s, the French mathematicians Pierre Fatou (left) and Gaston Julia pioneered the study of iterated functions that would later give rise to the Mandelbrot set.
Left (Fatou): Collection familiale. Right (Julia): Deutsches Museum, Munich, Archive, PR 01671/01
Introduction
The French Academy, and mathematicians more broadly, had another reason to be interested in iteration. The process played an important role in the study of dynamical systems — systems like the rotation of planets around the sun or the flow of a turbulent stream, systems that change over time according to some specified set of rules.
The prize inspired two mathematicians to develop an entirely new field of study.
First was Pierre Fatou, who in another life might have been a navy man (a family tradition), were it not for his poor health. He instead pursued a career in mathematics and astronomy, and by 1915 he’d already proved several major results in analysis. Then there was Gaston Julia, a promising young mathematician born in French-occupied Algeria whose studies were interrupted by World War I and his conscription into the French army. At the age of 22, after suffering a severe injury shortly after beginning his service — he would wear a leather strap across his face for the rest of his life, after doctors were unable to repair the damage — he returned to mathematics, doing some of the work he would submit for the Academy prize from a hospital bed.
The prize motivated both Fatou and Julia to study what happens when you iterate functions. They worked independently, but ended up making very similar discoveries. There was so much overlap in their results that even now, it’s not always clear how to assign credit. (Julia was more outgoing, and therefore received more attention. He ended up winning the prize; Fatou didn’t even apply.) Due to this work, the two are now considered the founders of the field of complex dynamics.
“Complex,” because Fatou and Julia iterated functions of complex numbers — numbers that combine a familiar real number with a so-called imaginary number (a multiple of i, the symbol mathematicians use to denote the square root of −1). While real numbers can be laid out as points on a line, complex numbers are visualized as points on a plane, like so:
Merrill Sherman/Quanta Magazine
Merrill Sherman/Quanta Magazine
Introduction
Fatou and Julia found that iterating even simple complex functions (not a paradox in the realm of mathematics!) could lead to rich and complicated behavior, depending on your starting point. They began to document these behaviors, and to represent them geometrically.
But then their work faded into obscurity for half a century. “People didn’t even know what to look for. They were limited on what questions to even ask,” said Artur Avila, a professor at the University of Zurich.
This changed when computer graphics came of age in the 1970s.
By then, the mathematician Benoît Mandelbrot had gained a reputation as an academic dilettante. He’d dabbled in many different fields, from economics to astronomy, all while working at IBM’s research center north of New York City. When he was appointed an IBM fellow in 1974, he had even more freedom to pursue independent projects. He decided to apply the center’s considerable computing power to bringing complex dynamics out of hibernation.
At first, Mandelbrot used the computers to generate the kinds of shapes that Fatou and Julia had studied. The images encoded information about when a starting point, when iterated, would escape to infinity, and when it would become trapped in some other pattern. Fatou and Julia’s drawings from 60 years earlier had looked like clusters of circles and triangles — but the computer-generated images that Mandelbrot made looked like dragons and butterflies, rabbits and cathedrals and heads of cauliflower, sometimes even disconnected clouds of dust. By then, Mandelbrot had already coined the word “fractal” for shapes that looked similar at different scales; the word evoked the notion of a new kind of geometry — something fragmented, fractional or broken.
The images appearing on his computer screen — today known as Julia sets — were some of the most beautiful and complicated examples of fractals that Mandelbrot had ever seen.
Merrill Sherman/Quanta Magazine
Merrill Sherman/Quanta Magazine
Introduction
Fatou and Julia’s work had focused on the geometry and dynamics of each of these sets (and their corresponding functions) individually. But computers gave Mandelbrot a way to think about an entire family of functions at once. He could encode all of them in the image that would come to bear his name, though it remains a matter of debate whether he was actually the first to discover it.
The Mandelbrot set deals with the simplest equations that still do something interesting when iterated. These are quadratic functions of the form f(z)=z2 + c. Fix a value of c — it can be any complex number. If you iterate the equation starting with z=0 and find that the numbers that you generate remain small (or bounded, as mathematicians say), then c is in the Mandelbrot set. If, on the other hand, you iterate and find that eventually your numbers start growing toward infinity, then c is not in the Mandelbrot set.
It’s straightforward to show that values of c close to zero are in the set. And it’s similarly straightforward to show that big values of c aren’t. But complex numbers live up to their name: The set’s boundary is magnificently intricate. There is no obvious reason that changing c by tiny amounts should cause you to keep crossing the boundary, but as you zoom in on it, endless amounts of detail appear.
What’s more, the Mandelbrot set acts like a map of Julia sets, as can be seen in the interactive figure below. Choose a value of c in the Mandelbrot set. The corresponding Julia set will be connected. But if you leave the Mandelbrot set, then the corresponding Julia set will be disconnected dust.
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