Bifurcation (2004)
Change can occur in varying degrees of violence and unpredictability. Accordingly, there are different kinds of mathematics for each.
At the mildest extreme lie the orderly changes of a system obeying differential equations, gliding along according to laws of motion. Think of the planets orbiting the sun or the burbling of a brook. Here calculus is in its element. It is designed for systems that evolve according to definite rules.
At the opposite extreme lies wild, irrational change. The kind that comes from senseless shocks to a system, external events bearing no logical relation to the system itself. The asteroid that hit the earth and wiped out the dinosaurs would be an example. The logic of the earth’s ecosystem instantly became irrelevant. Calculus can make no sense of such situations. No mathematics can. Probability theory tries its best, by enumerating the statistics of chance events, but its predictive power in any specific situation is limited.
Between these extremes lie systems that, somewhat paradoxically, follow rules containing the seeds of their own disruption. The potential for dramatic change lies within them but remains dormant. All it takes is a little nudge, typically an imperceptible stress, to push them over the brink. We speak of a tipping point, a phase transition, the straw that broke the camel’s back. Transformations like this can be surprising and logical at the same time.
Here the relevant mathematics is known as bifurcation theory. As the term suggests, a “bifurcation” is a splitting, a fork of possibilities. More precisely, a bifurcation is a qualitative change in a system’s dynamics as some parameter is varied continuously. The contrast here is meant to be striking: the conditions (the parameters) change continuously, yet the resulting behavior changes discontinuously. You gradually turn up the heat and nothing happens until you reach the bifurcation point. Then the pot begins to boil.
Joff’s letters kept coming, once a month, sometimes twice. It got to the point where I stopped opening them. They’d lie on my desk, off to the side, a growing pile of envelopes addressed in that familiar cursive. Until one day a letter arrived whose cursive looked ragged.
Dear Steve, |
Sat. January 17, 2004 |
Eek! I had a mild stroke Thurs. noon and lost all sensation in my right (writing) hand. Several hours later I managed to open and close my fingers and get some strength back into my grip, but, alas, no dexterity! X@%X! A one-handed piano player isn’t in demand, so I’ll miss my gig with our jazz quartet tomorrow.
Just after I had completed the ‘‘rehab’’ of my new, titanium knee, I’m challenged to get control of my individual fingers. It is coming along; my middle and 4th fingers are holding back my progress. Trying to play the piano is helping, but frustrating.
My winter kayaking should work O.K.—as soon as this cold spell passes. Right now the saltwater cove I launch in is frozen too thick to get through.
Last week, while I was paddling, the cold icy water surely was giving my boat the “slows.” I thought of Reynolds numbers and decided to revisit my physics book, then try to show that the complex fraction of units collapsed to a dimension-free number.
This led me to a pleasant retreat through viscosity, stress and strain in solids, then back to liquids, and Poiseuille. . . .
More physics encounters; this time with friction. I had discovered 8’ posts that I could use, in the golf course dump. I dragged the first two laboriously through the woods, which included an uphill stretch through a boulder ‘‘patch.’’ That night I had an idea to use my mountain bike, which I use to tow my kayak, to drag the posts the long way on the edge of the course, then through a level section of woods to our mini-barn. It took a couple of rides through 3" of snow to discover that the least friction came when most of the weight of the post was on my rear wheel and the tail of the post dragged in the snow.
It’s evening now. At midday I rode my bike to my kayak launch site to affirm that this cold spell had frozen my access to the open waters of the Sound. Attempting to ride a farm road to the beach, I crashed when I skidded on an icy ridge covered by snow. Luckily I went down on my (good) left side. I’m used to banging shoulders; Sue and my sons are wishing I was more the type to relax by the fire with a good book. This seems like a reasonable hope.
Good news today: my self-invented rehab exercises are yielding encouraging results with my piano playing. Writing this finds me trying to get a comfortable grip on the pen.
Actually I really am reading a fascinating well-researched book about Magellan’s circumnavigation (without charts or longitude) of the earth. My grade school teacher gave me no hints as to the hardships involved. At least on Shackleton’s trials in the Antarctic, he knew where he was. The Magellan book is Over the Edge of the World. I’m glad not to have been a sailor in Magellan’s fleet.
Hope all is well with you, Carole, Leah, and Joanna. Sue and I are looking forward to our usual escape from the Conn. winter, to her family home on Hanalei Bay, Kauai.
Best regards,
Joff
I didn’t write back to Joff about his stroke. Or call him either.
Maybe it was a kind of exhaustion. My dad had died a few months earlier, in October 2003. Watching him deteriorate was awful. So tiny in that hospital bed, his lips dry, always wanting more ice chips, his eyes unfocused, the humilation of the feeding tube. He’d never been a man of many words. My mother, whom he’d loved with every molecule in his body, had generated enough conversation for both of them.
I don’t know if that had anything to do with why I didn’t write back to Joff. But eventually I did contact him. It took something else to precipitate it.
In April 2004 my brother Ian, 57 years old, died suddenly one night. Same as my mom: stomach pain, emergency room, gone.
As soon as he heard about it, Joff sent me a note of condolence. Which underscored that I’d never done the same for him. It was time for me to change.
Dear Steve, |
Monday, April 19, 2004 |
I got a letter from Loomis with the news that your brother, Ian, had died. Sue and I are very sorry to hear this. My memory may be off, but I wondered if Ian had played JV football when Don Polkinghorne and I coached the team. The deep recesses of my mind have coach of the backfield, Don Polkinghorne, calling ‘‘Ian,’’ but this singular recall may not have been accurate. In any case, know that we Joffs have a lot of sympathy for your loss, and a loss to the Loomis family of graduates.
Retirement from Loomis hasn’t been a complete separation. In my dreams (NIGHTMARES?) I struggle to find my classroom as the time for the start bell to ring approaches! X@%X!
Please give our best regards to Carole, Leah,
and Joanna.
Joff
I called him the minute I finished reading his note.
“Hello?”
“Hi Joff, this is Steve Strogatz.”
“Oh, Steve.” His voice descended mournfully when he said my name.
After thanking him for his condolences about Ian, I steered the conversation to his stroke. It was light, he said. He’d been shoveling snow. After finishing, he’d come in and sat down. Ten minutes later he felt something pop, “like a hiccup.” His right hand fell asleep and did not wake up. Sue brought him to the doctor, who informed Joff he had an enlarged heart (“What do you expect after all those years of sports?” he chuckled in reply) and high blood pressure, and that his heart was a little leaky (“I never had a boat that didn’t leak”).
Then he changed the subject back to our usual math problems. Suppose you put 100 strands of spaghetti into a pot. If you tie the strands together at random until no loose ends remain, what’s the total number of loops you expect to form? Laughing again, he said this question, posed by another of his former students, had been frustrating the hell out of him.
At the end of the conversation, something made me ask if I could visit him at his home for an afternoon, maybe sometime in the summer. We always visit Carole’s mother in the city in August, I said, and it would be an easy drive up and back. Joff was tickled by the idea.
On the morning of August 17, I drove north out of New York City, heading up Route 95 toward Old Lyme with a pocket tape recorder and a sense of trepidation. A few days earlier, I’d asked Joff if we could talk about some personal things we’d never discussed. He agreed, a bit reluctantly it seemed, but yes, he said, okay.
There were so many things I wanted to ask about but had always avoided. Jeff’s cancer. Marshall’s death.
I made a conscious effort to prepare my senses. Whatever was going to happen, I wanted to take it all in. I suppose it’s pretty obvious by now that I’d lived much of my life in my head. Most of it, really. But on this day, I told myself, I’m going to open my eyes and see Joff for the first time, and listen to him.
After driving for a little over two hours, I got off the highway at Exit 70 and navigated the back roads to Joff’s street. It was a dead end, with five houses on the left, all perched on a small hill. On the right was a marsh, with dozens of birdhouses standing on wooden poles poking out of the wetlands. Each one was numbered, with addresses like 
, π, and other silly constants.
I parked at street level. As I walked up the driveway, I heard the soft sound of a piano from inside the house. The screen door was ajar. “Hello?” The piano stopped instantly. Sue and Joff greeted me, both looking tan and youthful at75. I kissed her on the cheek. Joff and I hugged, gripping each other on the backs of our shoulders.
Sue took me on a tour of the house. I noticed a wall filled with photos and felt my breath catch in my throat. Even though I didn’t look at them, I felt myself starting to choke up—I sensed they were pictures of their three sons and their families. Sue took me upstairs to her studio to look at her paintings, mostly still lifes. On the way back downstairs, I went past more walls full of family pictures, and didn’t look.
Joff took me down to his basement workshop. Canoes, kayaks, and windsurfing boards hung on the back wall above cinder blocks on the floor. The smell—oil? mustiness?—reminded me of workshops from my childhood. He said he only has simple tools: a band saw, drill press, another kind of saw, but not a rotary saw. “Why do I need a rotary saw?” he asked with a smile. He said he was proud of the windows that overlook the marsh and let in a lot of light. He told me stories about the numbering, and about his excursions on kayak to repair the nest boxes when the marsh is flooded high enough in the spring.
Our next destination was the porch deck. We sat under a big umbrella and had lunch: cold cuts and various breads followed by cherry pie with vanilla ice cream.
When it was finally time for our conversation, I took out the tape recorder. Joff began leafing slowly through his personal journal, which included math problems inspired by his outdoor activities. We went through it page by page, looking at his drawings of birds seen at this home or in Hawaii. Then lots of stories about his friend Hank, the bird guru.
I wonder, will this go on and on? I fight with myself, about my own impatience. This matters to him, so pay attention. I try to.
Eventually we got talking about Jeff, who had grown up to become an engineer. He had had testicular cancer in his early twenties, battled through surgeries and chemo, and made a great recovery. Afterward he appeared on television, snowboarding down a mountain for a Channel 3 newscast, saying he beat testicular cancer. Hearing about his miraculous recovery made me feel let off the hook for my longstanding callousness.
After a lull, I asked about Marshall. I’d been waiting to find the right time. My words spilled out.
“So I don’t think we ever talked about Marshall, but I wanted . . . I . . . I didn’t really know him . . . but he . . . I know that he died very young, and I . . . what, what happened to him?”
“Well, we, you know, that’s something we don’t really. . . .”
“You don’t want to talk about that?”
“Ahh, well, it was a sad. . . . He, he had. . . .”
“I remember him as a star. . . .”
Our words were colliding now. I’d gone too far. Feeling frantic, I looked for a way out. Then, unexpectedly, Joff began.
“He had a wonderful 27 years. He went to Mannes. He started at Amherst but music was gonna be his thing. He was interested in concert piano. And so he went to Mannes. . . .”
“. . . Which is a music school?”
“Yeah. It’s near Juilliard. It’s in that complex of music schools. It’s not Juilliard but it’s what he could get into. And, oh, he loved it. He had a little apartment down there, and we used to go down and visit him. And, uh, then . . . oh, he got this illness which was the thing that ultimately did him in.”
We sat quietly.
“And, it was a sad . . . because even, even in his waning moments, he’d stay up all night long, playing the piano, and just . . . the house would be filled with beautiful music. And he had made plans to get a job at the New England Conservatory and things like that, but the fates were wrong for him. But at least, I don’t know anybody that had . . . more . . . interesting things in that 27 years. I mean, he starred in every musical production the school theater ever had at Loomis. He sang in Jim Rugen’s madrigals. He composed—I was going through the files the other day and he’d written this kyrie, a religious piece that had been recorded.”
“Did he have a religious feeling?” I asked.
“Mmm. . . .”
“Not really?”
“Mmm, yeah. I think he did. I think he felt close to having to come to terms with somebody out there.”
Joff fell silent. We sat quietly and looked out over the salt marsh.
“So, I guess, that was a good thing, that, that, I think he went peacefully. . . . Oh yeah, we miss him. He traveled all over. A friend of his said, ‘Hey, I just won tickets to Israel. You want to come? I’ve got two of them.’ So he was off there. He just . . .he had a great time.”
Neither of us spoke.
“He was in a boy choir, starting when his voice was little. And at the church in Hartford, he was the head boy in the choir, and their choir went over to Westminster Abbey and sang one summer. I’d always admire his discipline. You know, a lot of great things happened. He was quite a scholar. When Sue was studying for her master’s degree at Wesleyan, she’d call him up and ask him questions. He was in the business of researching some of these lesser known composers. The guy was just a storehouse of information. And I always envied him the trick that he’d be at home and we’d sit around the piano and he’d say, ‘Okay, what’ll we do?’ And I’d get out the Cole Porter songbook and just turn to a page, something that he’d never seen. He could sight-read it, play it, and sing it all at one time with us. And I thought, ‘God, this guy’s got a multichannel mind that I wish I had!’ “
I asked about one of Marshall’s friends I’d known, and before long the conversation returned to easier topics and calculus problems.
After a while Joff asked if I’d like to go to the beach for a swim, or maybe just stroll by the water. We found a nice patch of sand and discussed a calculus problem that one of his former students had sent him. It was a problem about waves and required a Fourier integral—a generalization of Fourier series with uncountably many sine waves—for its solution. This was a different kind of infinity than Joff had ever encountered before, a higher order of infinity. While I explained it to him, the sun began to set. We sat together on the beach and solved the problem, surrounded by the waves in Long Island Sound.
We made our way back to his house as the sky darkened. He and Sue fed me a quick dinner—leftovers from lunch—before my ride back to the city.
I kissed Sue and hugged Joff goodbye. As I was stepping out the door, he handed me an envelope with my address on it.
I climbed into the car and opened the letter. At the end of it, he’d written
It just dawned on me that I could save postage and hand-deliver this letter when you arrive. I confess being nervous about seeing you. Our friendship/correspondence has meant so much over the years. Just the thought of you braving the drive up 95 from New York City for a visit is overwhelming.
Best wishes for a safe passage!
Joff
He was nervous about seeing me. Unbelievable. I was nervous about seeing him. What was he nervous about?