Curious Geometries

Photo by Zeenat Mansoor.

On the first day of Math 190, Michael Frame shows the 80 students gathered in Dunham 220 a picture of a mountain. Thin and slightly stooped, Frame looks out at his class from behind large glasses and a scruffy white beard. Though he at first seems tired, his voice bounces as he begins to teach. He shows us three pictures — a child’s face, a brick wall, a snowflake — to represent three familiar types of symmetry. These are the sorts of shapes we might have cut out of folded construction paper in elementary school. The fourth kind of symmetry, and the business of this class, is fractal.

On the screen, the craggy red spires of the mountain loom high in late-afternoon light. We watch as Frame clicks to zoom in on a certain section of the mountain. The zoomed-in image echoes the form of the first, its spires made up of smaller spires. Each part of the mountain resembles the shape and texture of the whole, from the original peaks down to the eroded rivulets worn by rainwater in the side of the rock itself.

These patterns are everywhere in nature: the winding together of creeks into rivers, the complicated curving of a coastline, the branching of the veins in a leaf. Frame spends the rest of class whisking through a series of in-plain-sight fractals that range from bubbles in crepes to the distribution of certain words in the poetry of Wallace Stevens. Mathematical examples — a curve of infinite length, an infinitely subdivided line — have slipped in here and there, but the emphasis is on the relationship between math and art, math and science, math and the world around us.

“Now there is one thing you should know,” Frame announces in the last minute of class. “I have an inoperable tumor.” The word he uses — “inoperable” — is precise and ambiguous. He explains that he will likely be confused and tired at times. He intends to finish the semester; the implication is that he might not.

Michael Frame’s first memory is of looking at the sky and wondering about the clouds.

His later research into the mathematical properties of chaos would reach back up into the clouds to describe their turbulence. The clouds, the stars, the scattering of galaxies beyond that, all find their place in his life’s work.

Born on Bastille Day, 1951, in Spring Hill, West Virginia, Frame had a curiosity that stretched far beyond his own horizons. He made baking soda grenades out of old pill bottles, and he dismantled radios to see how they worked, endlessly exploring the scientific processes that run the world. “Sports were never a real possibility,” he jokes with a gesture of hopeless gawkiness. “I was just focused differently.”

And so he read. He read everything, from science books for children to works of literature — in high school, he would develop a taste for Plato and Dostoyevsky — to the sets of encyclopedias that his parents bought from the grocery store.

One day, Frame came across an encyclopedia entry on Portugal: history, language, cultural, and geographical statistics, including the length of its border with Spain. When he arrived at the entry on Spain — history, language, culture, geography, and the length of its border with Portugal — he was startled to realize that the two lengths did not match. “This was an epistemological crisis,” Frame remembers. “When you’re 12 years old, you think truth lives in the encyclopedia.”

In fact, both numbers were estimates based on different units of measurement. When measuring the precise turns of an erratically wiggly line, such as the natural boundary along a meandering river, only a very small unit of measurement can capture each tiny divergence from a straight line. If one surveying team measured every wandering meter of the Spain-Portugal border, while the other only measured using straight shots of a kilometer, the two answers might differ by hundreds of kilometers. Frame at 11, asking questions the encyclopedia couldn’t answer, realized this of the irregular world: the smaller the measuring stick, the longer the measure would be.

The truth lives with whoever asks the right question. Years later, Frame would come across this border problem again in an article called “How Long is the Coast of Britain?” by a mathematician named Benoit Mandelbrot.

Water molecules, when viewed under a microscope, move in chaotic patterns as the agitation of their atoms jostles them against one another. The path of each molecule follows a clean, predictable trajectory until another water molecule slams it in a new direction. In nature, the movement of individual water molecules can ultimately determine the shape of hurricanes. This is why weather forecasts are so unreliable — even the slightest variation at the beginning of a process can be amplified over time to create unforeseeable events.

These same chaotic forces form snowflakes, whose famously individual symmetries arise like perfect six-sided puzzle pieces amidst the turbulence of clouds and atoms. “The rules for building a fractal are the story of how it grows,” Michael Frame tells his class. Each snowflake would tell the story of its journey through the storm, if only we knew how to read it.

In March of 1980, French mathematician Benoit Mandelbrot first saw the picture of the mathematical object that catalyzes a new way of seeing the world.

As with most great discoveries, its elegance was in its simplicity. Produced by a one-line single-variable equation, the shape at low resolution looked like an arrangement of recognizable geometric shapes: the heart-shaped cardioid, the progression of quickly shrinking discs, the short spike of a line segment.

But Mandelbrot noticed strange specks scattered symmetrically along the periphery of the cardioid. Random flaws in the image quality, he knew, could not produce such regularly-spaced blips. At higher magnifications, these blips turned out to contain smaller copies of the whole shape, nestled among the swirls of the complex border. These were in turn made up of smaller copies of the whole set, and so on. The more he zoomed in, the further he saw.

This keystone discovery sparked the imagination of scientists and laypeople alike. The tie-dye spirals of the Mandelbrot set’s complicated coastline flooded the popular consciousness. These images appeared on the covers of science and graphics magazines, on posters and t-shirts, and on screen savers of early personal computers.

Classical mathematics had always been full of strange examples that seemed to break the known laws of infinity. Until Mandelbrot put the pieces together, these were merely monsters, blips on the screen of the knowable universe. Mandelbrot made them visible.

In 1982, a detail from the Mandelbrot set appeared on the cover of Mandelbrot’s book, “The Fractal Geometry of Nature.” Two years later, a student at Union College brought the book to Frame, who decided to develop an introductory-level course on the material, which later led to an invitation to give the opening lecture on fractals at SUNY Albany’s commencement ceremonies, where Benoit Mandelbrot told him that one day they would work together.

In Tom Stoppard’s play “Arcadia,” one of Frame’s favorites, mathematics graduate student Valentine waxes poetic about natural chaos: “People were talking about the end of physics. Relativity and quantum looked as if they were going to clean out the whole problem between them. A theory of everything. But they only explained the very big and the very small. The universe, the elementary particles. The ordinary-sized stuff which is our lives, the things people write poetry about — clouds, daffodils, waterfalls, and what happens to a cup of coffee when the cream goes in — these things are full of mystery, as mysterious to us as the heavens were to the Greeks.”

The importance of fractals lies in the fact that nature, unlike clean-edged Euclidian geometry, is based on complexity. One can find the limit of a parabolic curve to determine its length, but the rough curve of a coastline weathered by storm and sea resists simplification. “Clouds are not spheres,” Mandelbrot wrote. “Mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

The forms of the natural world are not ruled by quadratic equations, but they are, nevertheless, ordered. The way a tree branches from its trunk to the tips of its twigs is regular; any segment of limb or branch resembles the structure of the whole. The genes that encode the instructions for building a tree contain no blueprints, only a small set of rules that will teach the tree how to build itself.

A few years after their initial meeting, Frame was surprised to receive a call from Mandelbrot. True to his word, Mandelbrot invited Frame to spend the next year working with him at Yale.

Frame was more than happy to collaborate with the founder of his field, though he did not understand why Mandelbrot had chosen him. “I’m not really very bright,” he thought, though he admits that he has a talent for explanation.

At Yale, Frame taught a version of his fractals class and worked with Mandelbrot on investigating fractal characteristics. Their first project involved measuring the distribution of the empty spaces inside complicated fractal structures. After outlining the general problem and the probable way through it, Mandelbrot gave Frame his prediction. Frame’s job was to prove it.

He labored at the calculations for two weeks. Intimidated by the accounts he had heard of Mandelbrot’s terrible ego, Frame checked and double-checked his work carefully. By the end, there could be no mistake that Mandelbrot’s predictions had been wrong.

Frame brought the results to Mandelbrot with not a little trepidation. To his surprise, the infamously arrogant mathematician accepted the proof with no sign of a bruised ego. “Marvelous!” Mandelbrot said. “The answer is much more interesting than I had thought.”

Frame tells the story in Mandelbrot’s thick French accent, then switches back to his own voice. “He was a scientist.” An answer that perplexed was as good as or better than an answer he could predict, because it led to the next set of mysteries. For Frame, this event showed that they could work together. And as Mandelbrot would eventually acknowledge, he had also decided then that the two of them would make a good team: Frame was not too afraid to tell him when he had done something wrong.

“Of course,” Frame admits twenty years later, “I really was terrified.”

Over time, Frame realized that the apparent arrogance for which Mandelbrot became known was a side effect of the attention he had received for the discovery of the Mandelbrot set. “For 20 years, he was the hermit crying alone in the wilderness,” Frame says. “After being ignored for so long, he was surprised to be taken up so quickly and so completely.” The sudden academic interest in fractals often overlooked the work Mandelbrot had already done ten or fifteen years earlier — and Mandelbrot was not averse to saying so.

Instead, Frame remembers Mandelbrot for his intense loyalty. If you worked closely with him, Frame remembers, “You were his folks. He would do anything he could to help you.”

After his sabbatical at Yale ended, Frame alternated between teaching at Yale in the fall and Union in the spring. This worked for a few years until Union objected. Mandelbrot, hearing this, said, “We’ll see about that.”

Not much later, Frame was offered a full-time position as adjunct professor at Yale. He took the job, even though it meant leaving the certainty of his tenured position. “If it hadn’t been for Benoit’s being here,” Frame told me, he would have stayed at Union. “Then again, if it hadn’t been for Benoit, I would never have come to Yale in the first place.”

That first year at Yale, Frame taught his introductory course on fractals. It was the first non-calculus course the math department had ever offered. Several well-meaning faculty members reassured Frame that he should be happy if even 20 students enrolled. On the first day, there were 180.

Since then, Frame has reinvented Yale’s calculus curriculum, designed special calculus courses that focus on applications in biology and medicine, and taught classes for non-math majors. The rules of logarithms and integrals are ultimately less important to Frame than being able to watch that “spark” going on in the eyes of his students. “That is truly wonderful,” Frame says. “There is no way to fake it, and there is no way to hide it when it is happening.”

“I guess under the circumstances, I should be gloomier than I am,” Frame tells me one day, sitting in his office with his hands folded across his lap. He fell down the day before on his way home from work; the therapy to treat his tumor sometimes makes him dizzy. Though he seemed shaken earlier, he now looks calm. “Curiosity is a wonderful thing. If anything will save us as a species, it’s curiosity.”

The second project that Frame and Mandelbrot worked on together dealt with a certain shape called a self-contacting fractal tree. In this structure, a line segment divides at certain intervals into two shorter segments until the “branches” began to curl back again and, in the exact middle, touch.

As they began, Mandelbrot again gave Frame the trigonometric formula that he expected to be the result of their work. This time, Frame’s proof came to the same answer. “This was complicated stuff,” Frame says of the equation, a matter of sines and tangents raised to unusual powers. And yet the precise geometric relationship between each variable had been immediately apparent to Mandelbrot. “I asked him how he did it and he said, ‘I can just see these things.’”

“In cases of genius, it is helpful to be able to see what ability sets one apart,” Frame says. “For Mandelbrot, it was his remarkable ability to visualize mathematical problems as geometrical shapes in his head.” His tic of rapidly forming associations between disparate objects was not always easy for other people to follow in conversation. “Benoit had certain odd linguistic traits,” Frame says. “Because he was so good at seeing connections, he could never stop seeing connections.”

This talent was ideal for synthesizing fractal geometry from fields as disparate as aeronautical engineering and art, geology, and astrophysics. “He read everything imaginable,” Frame says, remembering phone conversations that wandered through six or seven tangents of tangents. To keep track, Frame held up a finger for each new digression.

Frame’s understanding of Mandelbrot’s personal vocabulary was better than average, perhaps because the two of them shared the same fondness for digression. What in Frame is merely a tendency to wander was almost another language for Mandelbrot.

Frame remembers one instance of this confusion took place at Clark’s Dairy, where the two of them often had lunch together. Mandelbrot always ordered the same thing: a bowl of pea soup with rye bread. (Frame had a cheese sandwich.) When the waitress came to take their dessert orders, Mandelbrot asked for a bowl of butter pecan ice cream “with a funny hat.”

Frame was as confused as the waitress, until he realized that an upside-down ice cream cone would resemble a strange sort of dunce’s hat. To Mandelbrot, the connection was obvious. “He was not always straightforward,” Frame explains with a wry grin.

Mandelbrot frequently referred to himself simply as “a storyteller.” Frame’s job, then, was to make Mandelbrot’s ideas accessible to others: “To translate them from Benoit into human!”

Mandelbrot collaborated with people in many different fields, but these were all what Frame categorizes as “adult relationships.” With Frame, Mandelbrot was free to wander.

“By some miracle our internal clocks had stopped at the same time,” he explains. “In the image I keep returning to, he and I are two little kids, running around in a field under a big blue sky, showing each other new things.” He cannot explain the image — it has no counterpart — but he returns to it again and again. “It was just pure pleasure of discovery. There was no posturing.” There they were, for twenty years, pointing and sharing and wondering under a limitless expanse of sky.

Michael Frame’s library has been carefully subdivided with extra shelving into cramped aisles lined on both sides with hundreds of books. Among them are whole shelves devoted to Dylan Thomas, Wallace Stevens, Paul Auster, Scott Bradfield, Richard Powers, T. C. Boyle, Haruki Murakami, David Mitchell, Jorge Luis Borges, José Saramago, Marcel Proust, the criticism of Roger Ebert, a 25 volume set of Mark Twain, the complete Encyclopedia Britannica, and 60 volumes of an ambitious series called “Great Books” — a title that covers everything from Homer to “20th Century Imaginative Literature.”

“I think fiction tells us deep things about the world,” Frame tells me, coming in from the kitchen where he has been making crepes. “José Saramago had a huge influence on me.” His favorite book of Saramago’s is “Death with Interruptions,” which he now takes down from the shelf. “The last page and the first page taken together are the most perfect piece of writing I’ve ever read. The temptation to read them to you is almost overwhelming.” Staying himself, he puts the book back on the shelf.

Upstairs, Frame’s office holds volumes on chaos and fractals and bifurcation theory, among others. Books lie piled up on tables and on the floor, having already packed solid every inch of shelving. Among this impressive collection are various paintings and prints, some fractal, some not, many painted by his students or given to him by friends.

Another he shows me is the first piece of artwork he ever purchased: a small, square canvas of an abstract starburst pattern, bought for five dollars at a craft fair in West Virginia. The paint is layered on thickly in deep teals and blues that stand out in ridges from the canvas, though it has developed a network of large cracks over the years. Its effect on him was immediate, he says: “You could just get lost in it.”

In August last year, Mandelbrot called Frame. “I knew immediately something was wrong,” Frame remembers. “He was sobbing. ‘I won’t be here long.’” Mandelbrot had been diagnosed with pancreatic cancer, a notoriously aggressive disease with a very low survival rate. The doctors predicted that he had four to six weeks to live.

Mandelbrot only wanted to focus on the work left to be done. The two of them had been collaborating on two new books on fractals, as well as a paper on a particularly tricky problem that Frame would not be able to solve on his own. Mandelbrot had also been working to finish his memoirs. At the time of his diagnosis, multiple drafts existed in complicated states. The task of clarifying and cutting it down required months, not weeks.

“Benoit was heartbroken and furious that he wasn’t going to be able to finish the memoirs himself,” Frame remembers. They were meant to be “his victory lap,” his chance to tell the whole extraordinary story himself: from his childhood in Poland to hiding in the French countryside during the German occupation of France in WWII, to his discovery of the Mandelbrot set.

Frame finally convinced Mandelbrot to tell some colleagues and friends about his illness. Old acquaintances, colleagues in the department, people with whom he had collaborated, all sent letters and emails with words of gratitude and support. Many told him that fractals had revolutionized the way they navigated their fields; it was often more surprising not to find fractals.

Mandelbrot thanked Frame for having enabled this kind of gratitude. “I attempted to say something about how influential he had been,” Frame says, “but he already knew this anyway.”

When Mandelbrot died in October, two months after his diagnosis, he left behind the work that no one else knew how to finish. Mandelbrot’s widow and his assistant took on the task of editing the memoirs. Frame helped with the technical translation and wrote the afterword, the first draft of which he wrote out longhand. “That was really a hard thing to do,” Frame admits. “I filled up ten pages before I looked up.”

The book, called “The Fractalist,” will be out by the end of the year. “I’m embarrassed to say that there is a short section about me,” Frame adds with characteristic reticence. In fact, their partnership defined much of the last decades of Mandelbrot’s career. At the memorial service, a colleague remembered that the great mathematician had once said that his greatest success at Yale was bringing in Michael Frame.

A year later, Frame insists on a simpler view. “We really were little kids together. I truly miss that.”

I’m getting awfully tired of being old and sick,” Frame admits sometimes. Though his reaction to the experimental treatments has been better than expected, his health continues to decline. “In weaker moments, I get frustrated that I won’t see how things turn out. In that sense, I have nothing but envy for you young guys … You’ve got decades and decades.”

Frame’s grandmother saw the first flight at Kitty Hawk, and she saw the first man on the moon. In Frame’s lifetime, he has seen the invention of the Internet and the proliferation of the laptop. “That was mammals opposed to dinosaurs,” he says. “Nobody saw it coming.” He only wonders what can come next.

“It hit me a year ago that I don’t really care what happens anymore, at least to me. There are things I am interested in seeing,” Frame admits, but he is most aware of the great happiness he has found in his family and his work. “Ever since I began learning geometry, I’ve known that understanding how something works, how the pieces fit together and why, is a pleasure subtle beyond all common measure. I have gotten to spend the bulk of my life helping several thousand students have a few moments of this pleasure. For me, there is nothing better.”

Tegmark’s multiverse theory hypothesizes that a truly infinite universe would go on and on for so long that it would have to contain all possible arrangements of matter. Frame describes this to me one afternoon, gesturing toward the cinderblock walls of the office as though the infinite lay just beyond. “My own view is perpendicular to this: what if there were infinite time?”

In the cosmological short term, the galaxies of the visible universe are drifting farther and farther apart. Stars are burning off into white dwarves, or collapsing into supernovas that scatter heat and energy across the sky. Our sun will die some spectacular swollen death and the planets ringing its bonfire will eventually turn to ice. One by one the stars will go out, or maybe all at once — disappearing over the horizon faster than their light can get to us. After that, everything will be swallowed into black holes. Then, the ultimate darkness of a universe gone still.

And after that?

“If you wait long enough,” Frame says, a spark in his eye, “every state that is accessible will occur.” The black holes will evaporate particle by particle. The energy which had dissipated across the whole universe will at some moment reconvene by random chance. The whole universe will stutter back into life — it would have to, given infinite time to do so. “We will be back in this room again and again and again with every variation … Not just us, but the objects around us will return again and again in every configuration. The whole universe will just reassemble randomly.”

In the afterword to Mandelbrot’s memoirs, Frame writes a version of this story:

“My first memory is of looking at the sky and wondering about the clouds. I’ve been fascinated by ‘up’ ever since … A half-century ago, visiting my grandparents on a summer evening, my grandfather and I were lying on his driveway, warm from the day’s sun, watching the stars come out as the sky darkened. Darkened isn’t right, although that’s what happened … It isn’t that the evening sky is higher; it’s deeper. Watching blue to azure to violet to black, I had a moment of dizziness, when instead of looking up, I was looking down into impossible depths. The lonely space between the stars swallowed me, Gramp’s voice drifted away. Was the tightness in my chest fear? No, it was curiosity, sharpened by the hint of an amazing surprise just slightly out of reach. What was this sense of falling trying to show me? I could almost understand it. Then my grandfather’s hand on my shoulder and his question, ‘Did you fall asleep, buddy?’ I wanted to say, ‘No, Gramp, for the first time in my life I was just about to wake up.’ But I knew I couldn’t explain what I meant, because I didn’t know what I’d meant. More directly than you might think, the years I’ve spent studying math and physics, all the papers and books and software and web pages I’ve published, the thousands of students who’ve listened to me try to explain math, have been, in broad strokes, about trying to figure out what I’d have meant by ‘just about to wake up.’ This is the context of my interior life. If there is one, the answer to what I’d have meant will lie in the details.”

• sutherland4l

Beautiful article, Amelia! It was hard for me to read, though… The person next to me won’t stop cutting onions.

• Inigo_Montoya

Michael Frame is one of Yale’s greatest teachers. Thank you, Professor Frame, and thank you Amelia Urry, for writing a portrait worthy of him.

• EllenSong

This is so well-written! Amelia this was a wonderful portrait of two wonderful mathematicians.

I’m a math major and I see their “little kid mannerisms” all the time and it is the most CHARMING thing in the entire world. That little kids line really made me year up! And the curiosity.

Awesome, awesome article. Thank you.

• ROFLCOPTERSUX

Amelia (in no h)Urry, this is beyond beautiful.

• Leah

Thank you so much for writing this, Amelia Urry. Michael Frame was, without question, the most extraordinary professor I had at Yale. I came to Yale as a math enthusiast and took 290 my freshman year. Ever since, Frame is always the person I recommend to any friend who mistakenly thinks they don’t like math. He’s amazing at awakening people to the beauty of looking at the world through mathematics, but he’s my favorite because he’s so ready to share his enthusiasm about looking at the world, period.

• mlauter

I agree, this is a beautiful article. Michael Frame is the best teacher I have had at Yale, and he is a wonderful person, and you really captured that and more.

• SY

Fractal geometry should be a Yale online class so more than 5,000 students can take it. An online class also would be a good addition to the Mandelbrot memoirs.

• joematcha

Yes, SY, I absolutely agree. Fractal geometry was such a pleasure I would love to sit through it again online.

• Inigo_Montoya

Thirded. It would be a perfect Open Yale Course. If Frame is able to teach it again, Yale should definitely record it.

• Rachel_B

This is an excellent article – thank you. And Michael Frame is a wonderful teacher, with a remarkable ability to show students taking basic calculus courses the imaginative elements of mathematics. I think he is able to communicate in such a lovely way the beauty of mathematical precision and, in this way, to make people *want* to understand technical details.