All you really need to know about the mathematician Ronald Graham is that, with his 80th birthday fast approaching in October, he bought a ‘hoverboard’ — a two-wheel mini-scooter, with electronic motors operated by a pressure plate under each foot. “I couldn’t resist,” says Graham, who adds that he only fell once when he started practicing in the kitchen while his wife was away in Taiwan.
Which is to say that Graham is an irrepressible ringmaster of fun, whether it’s purely for play or exuberantly enlightening, in a circus land all his own.
Indeed, when Brady Haran, host of the high-voltage Numberphile YouTube channel, visited Graham to shoot some footage, Haran knew enough to bring along a large supply of brown postal wrap paper. It was for Graham’s expositional high-wire act illustrating a very, very big number that Graham discovered in 1971. Called Graham’s number, it seems in theory to be so unimaginably large that Donald Knuth, a computer scientist at Stanford University, invented notation using arrows as a short form for the exponential tower of augmentation that ensues when one tries to get ahold of Graham’s number.
3 ↑ 3 = 27
3 ↑↑ 3 = 7, 625, 597, 484, 987
(or, 3 to the 27th power)
3 ↑↑↑ 3 = 3.6 trillion digits or 1.258014298121 … x 103638334640024
(or, 3 raised to the 7,625,597,484,987th power)
While displaying this same example on Numberphile, Haran discovered that he did not have enough screen space. Soon after that, Graham says on camera, “I think we ran out of this piece of paper.”
“Want some more?” Haran asks.
“Yeah. You ain’t seen nothing yet.”
Behind the ‘Biggest Number’
When cameras and spotlights again turned up about a year later chez Graham — known among his friends as a man who accepts all invitations — it was filmmaker George Csicsery and his crew for this Science Lives series, along with Graham’s friend and fellow mathematician Joe Buhler of Reed College, who served as interviewer. Once again Graham’s number was on the agenda. (Fittingly, for his 80th birthday, his daughter made available lots of Graham’s number paraphernalia — mouse pads, postage stamps, purses, business cards, placemats, doormats, stickers, calendars and such.)
His namesake number is “kind of a joke,” Graham says. He happened on it while proving a combinatorial theorem using a technique called recursive multiple inductions. “When that happens, you tend to get large numbers,” Graham explains. “The problem was the following: Take an n-dimensional cube. Think about that; it has 2n vertices. You join any two vertices with a line segment. For example, take n = 3 — visualize a three-dimensional cube. There are eight vertices, which form 28 line segments. Now color each of the line segments red or blue, any way you like.
“The question is,” Graham asks, “can you always find four vertices so that all six of the line segments are the same color and these four points lie in a single plane? Will it always happen? Yes, it will always happen, if the dimension n is large enough. How large is large enough? Some really gigantic number.”
As described by Martin Gardner in his Mathematical Games column in the November 1977 issue of Scientific American, this number of dimensions seemed to be “so vast that it holds the record for the largest number ever used in a serious mathematical proof.” And it made the 1980 Guinness Book of World Records as the largest number ever published in a mathematical proof (although the so-called TREE sequence has since supplanted it).
By all appearances, Graham’s number should be a very big number, one that needs numerous arrows. But, counterintuitively, its true bound might not be so big after all. The problem is proving it. A 12-dimensional cube is not large enough for all six line segments to be the same color and lie in a plane, and provably so. But that is where certainty ceases. As Graham explains to Buhler (in chapter 23 of the video), the true answer to the problem — the upper bound for the number of dimensions— might in fact be a measly 13 or 14 dimensions. “Which just illustrates the gap between what seems to be true, and what we can prove to be true,” Graham says. “It’s a pretty big gap, and we’d like to close it down a little bit.”
The Mathematical Juggler
In addition to Graham’s number, another signature subject came up for discussion — and demonstration — during the Science Lives shoot: Graham’s famous juggling. In between takes, Graham and Buhler juggled clubs back and forth. At one point, Csicsery took a break from directorial duties and served as a prop: He stood between the jugglers and dangled a pencil from his mouth, and it occasionally got knocked out when the jugglers tested their acumen and deliberately threw a club ever so slightly off course.
As Graham has been known to say, juggling is a metaphor, and this applies to his life in general and to all of his myriad interests in particular. “Juggling is a metaphor for doing more things than you have hands to do,” he says. “If you only have two objects and two hands, how hard can that be? But if you have three or more objects and two hands, that’s pretty dynamic.”
He is often asked about the connection between juggling and math, and indeed the juggling world seems to be populated with more than its fair share of mathematicians and computer scientists. “It’s the philosophy of being in control,” Graham says. “The problem with juggling is, the balls go where you throw them, not where you wish they would go, or where they are supposed to go. If something goes wrong, it’s not the phases of the moon. You made a mistake. In fact, every throw is a little bit off, and you always have to be correcting for it.” Similarly, the problem with programming is that the computer does exactly what you tell it to do. “There is no command that says, ‘You know what I mean,'” Graham points out. “The computer doesn’t know what you mean, so you have the feeling of control. Mathematics is often called the science of patterns. In juggling you really are controlling these patterns in time and space. And in mathematics, you can never solve all the problems; it’s an unbounded challenge. In juggling you can’t do all the tricks. There’s always one more ball, or with your hands behind your back.”
Graham, it seems, is always game to add another ball — for instance, the hoverboard. Or the training for a half-marathon walk, another current project. How does he do it? “It’s called time packing,” he explains. “You gotta fit it all in. And then there’s math. Gotta do math!”
Certainly, he has done a lot of math. The citation for his Leroy P. Steele Prize for lifetime achievement, awarded by the American Mathematical Society in 2003, lauded him as “one of the principal architects of the rapid development worldwide of discrete mathematics in recent years” (pdf). And the distinguished mathematician and philosopher Gian-Carlo Rota, in nominating Graham for the presidency of the AMS in 1981, described him as ranking among the most “charismatic figures in contemporary mathematics” and “the leading problem-solver of his generation.” Graham has published five books and hundreds of papers with more than 200 collaborators, most frequently his wife, Fan Chung (their tally is 94) — and this is on top of his commitment to administration and teaching.
Graham’s propensity for hypertasking manifested early. After taking college courses in electrical engineering and mathematics for four years without earning a degree, he enlisted in the U.S. Air Force. He spent three years stationed in Alaska, where he moonlighted with more undergraduate studies and earned a degree in physics. He obtained his doctorate in 1962 at the University of California, Berkeley, with the number theorist D.H. Lehmer as his adviser. Lehmer was disappointed when Graham left academia and took a job as a researcher at Bell Labs, based in Murray Hill, N.J. But Graham quickly rose through the managerial ranks, serving as director of information sciences from 1962 to 1995, all the while enjoying several stints back in academia — at Princeton, Stanford and Rutgers universities; the California Institute of Technology; and other institutions.
It was at Bell Labs that the Stanford mathematician and statistician Persi Diaconis first met Graham, in 1978. Interviewing for a job, Diaconis walked into Graham’s office and found him juggling, standing in the middle of a net attached to the four corners of the room and tied at his waist. He had rigged up this contraption to avoid having to pause and bend over to pick up errant balls; instead, when he dropped a ball it would fall into the net and roll back to him, and Graham could resume almost without interruption. Diaconis recalls, “I thought, ‘Oh, this is going to be interesting.’ And it was.”
They’ve since written 21 papers together and recently co-authored the book Magical Mathematics. Fifteen years in the making, it won the 2013 Euler Book Prize and motivated Graham to compose a new talk, “Juggling Mathematics and Magic,” which he delivered, using 100 decks of cards, to an audience of 800 at the opening of the International Congress of Chinese Mathematicians in Taiwan in 2013.
Accessible Person, Original Thinker
According to Diaconis, Graham is so accessible and down-to-earth, and so much fun, that people sometimes do not realize that he is also a brilliant mathematician. “He is very, very original, and very, very clever,” Diaconis says — and clever in both the pure and the applied realms. At Bell Labs, Graham ensured that telephone calls were routed at maximum efficiency. But he has also built purely abstract objects from the ground up, eschewing advanced mathematical tools and technical machinery, especially in his field of Ramsey theory (Graham’s number originated from a problem in Ramsey theory), which, as Diaconis puts it, “really has nothing to do with the real world. It’s just beautiful, pure mathematics.”
Beyond his contributions to mathematics, Graham applied the leadership skills he honed at Bell Labs to become an active member of the mathematical and scientific community at large. He is currently chief scientist at the California Institute for Telecommunications and Information Technology in La Jolla. (One of his jobs there is to ‘live in the future,’ so he was soon due to go to Seattle for a medical checkup and forecast based on the sequencing of his genome and microbiome.) He served as president of the AMS from 1993 to 1994 and of the Mathematical Association of America from 2003 to 2005. Currently he is chairman of the Friends of the International Mathematics Union, and he recently squeezed in a layover meeting with the IMU’s president and treasurer at Chicago O’Hare airport, en route home from Taiwan. Why does he put in so much time? He responds that he feels an obligation to the profession; he is driven to make it appealing to as many people as possible.
“Graham has a knack of presenting real mathematics in a way that appeals to laymen, even if that was not the initial intent of his work,” says Terence Tao of the University of California, Los Angeles. And similarly, if a colleague needs an explanation, Graham is there to help. As Diaconis puts it, “I’ve had it happen that I was stuck on something and giving a lecture and called him an hour before — I’m in terrible trouble, I don’t understand the proof of something he’d outlined. He just puts everything aside and explains it to me in English. And that’s very valuable. You can count on him.”
“And it’s fun,” Diaconis adds. “It’s never not fun.”
It always comes back to fun: As well as juggling all these responsibilities — the private and public spheres of being a mathematician — Graham also juggles himself. That’s how he thinks of his trampolining hobby: Even as a newly minted octogenarian, he still bounces on his trampoline at least once a week, seldom strapping himself into the bungee cords he added for safety. But near the trampoline he has a slack wire, and while mastering the strange balance needed to walk the wire, he took to wearing a bicycle helmet, even though he placed mats beneath the wire. “It’s like I say often, ‘Good judgment comes from experience, and experience comes from bad judgment,” he observes. “So there’s a two-step algorithm to learn how to get good judgment.” (He figures the wrist fracture he suffered while practicing on his hoverboard will heal in about six weeks.) He also continues to do handstands in his gym (he converted one of his two garages). And he still delights in explaining why one-handed handstands — at which he excelled in his youth as a gymnast — are easier on a moving pedestal than on a fixed base like the floor: because acrobats can move the point of support rather than their center of gravity. “It’s the philosophy of the circus,” he states. “You want to do things that look hard but are easy, not things that look easy but are hard.”
Sometimes, on a really good day, all this happens at once. In The Man Who Loved Only Numbers, a biography of Graham’s great friend Paul Erdős, author Paul Hoffman describes Graham this way: “While Erdős could sit for hours, Graham is always moving. In the middle of solving a mathematical problem he’ll spring into a handstand, grab stray objects and juggle, or jump up and down on the super-springy pogo stick he keeps in his office. ‘You can do mathematics anywhere,’ Graham said. ‘I once had a flash of insight into a stubborn problem in the middle of a back somersault with a triple twist on my trampoline.’”
Life With Erdős
Graham’s kinship and collaboration with Erdős, who died in 1996, is one of the great 20th-century mathematical legends. They wrote 32 papers together, the latest of which was published in December 2015 in the journal Integers, titled “Egyptian Fractions With Each Denominator Having Three Distinct Prime Divisors” (pdf). Also co-authored with Steve Butler of Iowa State University, the paper has its roots in research that began more than 40 years ago.
Erdős and Graham first met in 1963. Graham set up an “Erdős room” in his house to accommodate his friend’s itinerant ways, and he also became Erdős’ banker of sorts, collecting checks earned as payment for talks and doling out prize money. “Erdős liked to offer small prizes for mathematical problems,” Graham says. “And for more serious problems, a bigger prize. He offered $10,000 for the problem about the large gaps between consecutive primes. And he said it was a rash offer.”
Although the prize was offer more than 50 years ago, this prime-number problem was finally solved by James Maynard in 2013 and, independently of Maynard, by Kevin Ford, Ben Green, Sergei Konyagin and Tao in 2014. “So I’m now on the hook,” Graham says. Tao, for his part, told Graham that he plans to pay the prize money forward. “There is a new bound on a gap between consecutive primes that has an extra triple log factor with a constant, and Terry said, ‘I’m going to offer $10,000 if anybody can prove that that constant goes to infinity.’ So it’s kind of an iterated set of problems.”
In Graham’s estimation, Erdős, with his endless repertoire of problems, had vision; in the back of his mind he had a bigger view of where mathematics was going. “I think Erdős had the idea that the problem sits in a larger context,” Graham says. “So many of his harmless-looking problems grew into something much larger. You never know when you plant a seed, a little acorn, will it just turn into a weed, or will it be a great oak tree? You don’t know.”
We’ll have a chance to find out, Graham notes, because not too long ago Tao struck again and solved the “Erdős discrepancy problem,” one of Erdős’ favorite problems. Tao proved not only Erdős’ specific conjecture but also the more general setting. “Where that will finally lead is hard to say,” Graham says.
One place it leads to is the question in mathematics as to what is more important: coming up with a conjecture, or devising the proof. “Sometimes,” Graham says, “if you know what you’re trying to prove, that’s half the battle. You often don’t know.”
The Illusion of Knowledge
And therein, Graham believes, lies a message. “When you’re trying to prove something, you’re kind of convinced it’s going one way, but it’s really going the other way. So I have a saying I always keep in mind. The main obstacle to progress is not ignorance, but the illusion of knowledge. You think you know. But no, you don’t. Once you understand that you don’t know, then your mind is a little more open to say, ‘Oh, OK, there are other possibilities, maybe it’s not true after all.’ Even though you wanted it to be true.”
And so it is with Graham’s own number — that upper bound on the number of dimensions that a cube must have in order to force the prescribed configuration of lines between four vertices. Which way is it going? Is it 13, or is it well beyond 3 ↑↑↑ 3?
It doesn’t work in 12 dimensions. “Twelve is not enough,” noted Haran in his Numberphile interview with Graham. “But it could be 13 dimensions?”
“It could be, yeah,” Graham replied.
“If you think you understand it,” Haran advised, “you probably don’t.”
After the Numberphile recording session, Haran recounts how he and Graham went out for sushi. Haran had turned his camera off, but Graham’s was still on. “He spent the whole time drawing and writing things on napkins,” Haran says. “I wish I’d been recording. He must have gone through 10 napkins.”
Afterward, Graham dropped off Haran at his hotel. As Graham drove away into the darkness — with his license plate reading NUMBER — Haran turned to the hotel concierges and said, “See that guy in the car? He invented the biggest number ever.”
They all looked wide-eyed and said, “Whoa.”