When the mathematician Michael Frame was growing up in St. Albans, West Virginia, he and his little brother Steven decided to make a hot air balloon. They didn’t have the money to buy a ready-made set, so Michael crafted a little bowl out of some tin foil and Steven found a bottle of rubbing alcohol to use as fuel. Together, they coated the inside of the bowl with sawdust so the alcohol wouldn’t slip out and attached a dry cleaning bag as the balloon. They struck a match and dropped it in and watched as the contraption rose into the night sky. Pretty soon the cup vanished from view; the only visible part was the bag itself, quivering in the breeze and reflecting the light from the flame below. The flight only lasted a few minutes, but it was enough for several people in the surrounding area to report UFO sightings the next day.
“Of course, they couldn’t tell the scale, so they saw our little dry cleaning bag floating in the sky and thought it was aliens,” Frame says with a laugh. Nowadays, Michael Frame’s life is all about scale. As a Professor Adjunct in the Mathematics Department at Yale University, a position he has held since 1992, he studies fractals—patterns that look the same close up as they do far away.
The most easily understood fractal shape is the Sierpinski Gasket. There are many different methods of producing the gasket, but the simplest way to visualize it is to start with a triangle. If you join the midpoints of each of the triangle’s sides, you’ll form a smaller triangle in the center. Remove that, and you end up with three smaller triangles surrounding an empty, triangle-shaped space. Repeating the process for each of these triangles will yield smaller and smaller triangles, each of which is part of a larger shape, which is in turn part of a larger shape. In the theoretical plane, this can go on forever, but comparable processes happen all the time. Take the bronchi in your lungs, each of which branches off into bronchioles which branch off into bronchioles which branch off into bronchioles. Similarly, if you look at a coastline, you’ll find that the erosion patterns that form within the stretch of a few feet are analogous to the same patterns that occur over a number of miles. It’s the same process repeated on smaller and smaller scales. The genius of it is that the range of its application seems limitless, from seashell design to planetary formation.
I once asked Frame, “Isn’t it overwhelming? I mean, it’s everywhere!”
He frowned. After a moment, he said, “Well, no. It’s all just a matter of finding a progression of simpler things.”
This is a common theme in how Frame sees the world. Someone once pointed to a sunflower and said to him, “How can you see that and not believe in God?” He said, “Easy. I just believe in Fibonacci numbers.”
I say, “What about things that can’t be explained systematically?”
He says, “I don’t think there’s anything that can’t be explained systematically.”
In addition to working for almost twenty years with Benoit Mandelbrot, the French mathematician who discovered fractals, Frame also teaches MATH 190, an introductory course in the subject that’s frequently taken by students outside the mathematics department because the study of fractals is as visual as it is analytic.
Mandelbrot discovered fractals while working as a probabilist for IBM in the 1960s. He was researching ways to predict and prevent problems in the way computers transmit information, and noticed that errors tended to clump together in patterns that looked the same whether he analyzed a shorter or longer time interval. In other words, the data behaved the same on a large scale as it did on much smaller ones. He called this property self-similarity.
Artists had been observing the property in nature for centuries (think of the wavelets within wavelets in Hokusai’s “Great Wave Off Kanagawa”), and the concept had been used to construct visual solutions to numerous mathematical puzzles, but no one had ever come up with a comprehensive theory to describe these processes in numerical terms. Mandelbrot had been interested in self-similar structures since the early ’50s, but the transmission error problem provided him with the right conditions to formulate a firmly grounded theory. By 1975, he had conducted enough research to publish Les objets fractal, an explanation of how any self-similar pattern could be described and replicated using only a few metrics.
Frame believes that there are patterns waiting to be recognized almost everywhere. If there doesn’t appear to be one, it’s probably because we’re not asking the right questions. A lot of that is probably due to the amount of time he spent working with Mandelbrot. Often, Mandelbrot, who died last October of pancreatic cancer, would write down a solution to a problem on a piece of paper and say something like, “These are the right answers, but can you figure out a way to explain them?” Frame would take them into his office for a day or two, working backwards to figure out how Mandelbrot had come to his conclusions. Mandelbrot told him once that equations came to him in visions. Frame’s job for a long time was to translate those visions into terms the rest of us could understand.
I ask him if he has had any eureka moments himself. He says, “I think eureka moments are for people with forceful personalities.”
Frame’s process is slow and deliberate, a fact about which he is mercilessly self-deprecating. Mandelbrot could look at an image or a problem and find a solution within a matter of seconds. When Frame wants to solve a problem, he’ll write it down and leave it on his desk, visiting in for a couple of minutes every now and then to squint at it until something clicks. He often complains he’s the dumbest person in the department. He’s proud of his teaching, but only insofar as he can empathize with students’ bewilderment. The trick to good teaching, he says, is “to remember what it’s like to not understand something. I’m confused about most things most of the time, so I recognize that look on students’ faces.” Despite this, he has worked as a consultant for the Department of Energy and the Department of Transportation and is in the process of designing a new set of classes for pre-meds that more accurately reflect real problems people face in the medical profession. For the past twenty years, he’s been working on writing a textbook for advanced fractal study.
The computer in Frame’s office is equipped with software that analyzes radio frequencies for signs of extraterrestrial life. Every few days, the data is processed and sent to the headquarters of the Search for Extra Terrestrial Life Institute. He knows he probably won’t find anything, but he thinks the idea is “dag-blasted cool.” The rest of the office is what you’d expect from a math professor: piles of loose papers, bulging manila envelopes, a board covered with swirling patterns of dust from lots of erasing. A cane hangs from the chalk ledge. Frame has had to use it more and more in the past few years as his strength has deteriorated. He is small, and he looks smaller when he’s seated behind his huge desktop monitor. His hair is white, receding, and closely cropped. An unruly beard of multidirectional hairs extends down his neck. His glasses are big and thick; they make him look a little like WALL-E, the trash-compacting robot from the Disney-Pixar film.
Frame was born in 1951 to Walter and Mary Arrowood Frame. Even though she made more money than Walter did, Mary quit her job as an executive at a local bank so she could stay home with Michael and his younger siblings, Steven and Linda. Walter was a gifted millwright. He loved his job because he loved using his hands, but he often had to work long shifts at odd hours. Every third week he would work the night shift, and everyone in the house would have to be very quiet during the day so he could sleep. Frame remembers sitting outside on the porch wrapped in a blanket while a thunderstorm raged overhead, waiting for his dad to come home. Every night, Walter would peel apples for the family while Mary read stories from a Hans Christian Andersen book.
At the time, St. Albans was just on the edge of rural West Virginia, and finding things to do was a challenge. (“A lot of kids were athletes, but obviously not me,” Frame says with a smirk.) Michael became interested in the natural world. He made hot air balloons and mixed homemade gunpowder with moonshine from his grandfather’s general store to create model rockets. One year for his birthday, his uncle gave him a telescope. He still remembers being able to make out the craters on the moon and, just barely, the rings of Saturn.
Michael was the first member of his family to go to college. His parents couldn’t afford the tuition, but the Frames have a history of doing what’s important first and worrying about logistics later. Walter himself ran away from home at sixteen to join the Navy. He lied to the recruiter about his age and started working as a repairman in the Pacific fleet, where he discovered his love for machinery. Frame remembers that soon after he was accepted into Union College, his father told him he was going to accept the foreman’s job he had repeatedly refused in the past because the extra money could help pay for his tuition. He tried fruitlessly for a while to talk his father into keeping his job as a machinist. Finally, Michael reminded Walter of a story Walter had once told him about Walter’s father from back when the Frames lived on a farm in Rosedale, West Virginia. One morning, Walter’s father went out to the corncrib to discover that someone had snuck in and stolen some of the chicken feed. He immediately threw out the lock to the crib. When Walter asked why, he said, “If someone is desperate enough to sneak into our corncrib so he can feed his family, I’m not going to stop him.” Michael told Walter that he would be worse off knowing that his father wasn’t doing what he needed to be happy than he would be with a little more money for tuition; he promised that he’d get by.
Frame went to Union expecting to major in physics, but when it came time to complete a lab requirement, he was dismayed to find that the only project available for an undergraduate of his experience level was “something in biophysics with killing frogs.” He has been a vegetarian for as long as he can remember (“Burned animals’ muscles—who thinks that’s a good thing to eat? I mean, come on!”), so he switched his major to mathematics. He had already completed most of the requirements anyway. From there, he moved on to complete his Ph.D. in mathematics at Tulane in 1978 and performed some postdoctoral research at Carleton University in Ottowa. The only thing he liked about Carleton was a hematologist who worked at the medical school named Jean Maatta, so he married her and they both hit the road for the New College in Florida. But they didn’t like the mosquitoes and moved to Schenectady soon after, where Frame was offered a teaching position at his alma mater.
He taught calculus and other basic math courses until 1988, when SUNY Albany announced that they would be bestowing Benoit Mandelbrot with an honorary degree. Jim Corbett, a professor at Albany, planned to deliver a brief lecture about fractal basics in advance of the ceremony. He couldn’t make it, so Frame filled in at the last minute. What he didn’t know was that Mandelbrot himself would be in attendance. He describes the experience as “giving a general introduction to the ten commandments with Moses in the front row.”
He must have done a pretty good job, because two years later, when Mandelbrot was invited to teach fractal geometry at Yale, he enlisted Frame to help him write the curriculum. The first time the course was offered in 1992 was just before the movie “Jurassic Park” came out. The film had been adapted from a book that ostensibly had to do with fractals and chaos theory, and everyone was excited about the prospect of understanding the science behind it. In reality, the book had very little to do with chaos theory and the movie had even less, aside from a few lines delivered by Jeff Goldblum about how nature always returns to disorder. That’s not really what chaos theory is, but 180 students signed up for the class anyway. Frame has taught it ever since.
Chaos theory, another mathematical concept that can be explained through fractals, is probably more applicable to Frame’s life than it is to dinosaurs. The most important aspect of chaos, and the one most often misunderstood, is sensitivity to initial conditions. The long-term future of a chaotic system is impossible to predict because even imperceptible differences in its initial conditions can lead to vastly different outcomes. In spatial terms, this means that you can never be sure whether your starting place is a specific point or another point arbitrarily close to that point: the two are indistinguishable, yet their futures are completely distinct from one another. This makes predicting the progress of the system as a whole impossible. The common example is of a butterfly flapping its wings in Africa causing a hurricane in North America. A more accurate way to put it, according to Frame, is that a butterfly flapping its wings in Africa makes it impossible to predict how, when, or where a hurricane will occur in North America because every small disturbance in wind patterns has the potential to yield a drastically different future.
“Chaos has given me a real appreciation for unpredictability,” Frame says. Who knows what would have happened had Jim Corbett not been busy on that day in 1988, or if the New College had been in a less swampy part of Florida, or if Michael Frame’s genetic material had come together in a way that made him more naturally fit for athletic activity? Each moment is a composite of untraceable forces leading back to an indiscernible initial position. In the case of the Frames, some forces are clearer than others. Michael had a melanoma removed from his ankle in 1983. His sister has had breast cancer since 1991, and his brother was diagnosed with leukemia in 2000. Two years ago, Michael was diagnosed with inoperable prostate cancer. He thinks unlucky genetics and continued exposure to Union Carbide chemicals account for the prevalence of sickness in his family. Cancer has also been one of the reasons why he and Jean—whom he calls the only person with whom he’s ever had an emotional connection—are so close. She has had breast cancer for twenty-eight years, and they both agreed early on that they wouldn’t have any children. He says, “The thought of having a child and then one or both of us dying while he or she was still young was just terrible.”
With regard to himself and his siblings, he says, “None of us have the sense that this thing is unfair.” He adds, “These are just the initial conditions of our lives.” For now, things are stable. He’s lasted longer than the doctors thought he would, and he’s currently on a few experimental treatments. He has good days and bad days. On the weekends, Jean, who now has a job in the transplantation center of Yale-New Haven Hospital, works in her vegetable garden. Frame watches her through his window or reads books by José Saramago, his favorite author. His favorite Saramago so far is Death With Interruptions, a surreal novel in which Death takes a vacation.
One of Frame’s favorite examples of self-similarity in art, Salvador Dali’s “Face of War,” also has to do with death. In the painting, a corpse-like face in a barren landscape looks out at the viewer. A skull sits in its mouth and in each of its eye sockets. Inside each of the skulls’ eye sockets and mouths is a skull with skulls in its eye sockets and mouth.
One day while sitting in his office, he explained to me why he likes it so much. “At some point, you can no longer see what’s going on in the eyes and mouth,” he said. “But you can imagine it.”