Bryant also played a role in the droll story of how passwords came to Yorktown. Yes, there was a time when our computers required no password! My older son’s math teacher was himself learning to program, and he introduced programming to his students. Having failed to make a certain program work, he asked my son to consult an expert—me! Begged to help, Bryant entered my account, wrote the program in no time, and printed a letter-size sheet for my son and his teacher.
A few months later, our computing center manager stopped me in the hall. “I am amazed. Of the substantial computer time available in the research division, you alone are using about half. I thought you were a very theoretical person.” “I am equally amazed because weeks ago I gave up programming for myself.” “So how come you are such a big user?” Monitoring revealed that I was billed for a mass of tiny programs run by high school students all over the surrounding Westchester County. At least one ingenious student or teacher had realized that simply typing my name in a box connected the user to the day’s biggest computer—at no charge.
At that point, the computing center staff had to assign passwords. So I can boast—if that’s the right term—of having been at the origin of the police intrusion that this change represented. Of course passwords must have originated in many places, and IBM Research’s turn would have come very soon.
Computer Graphics Before IBM Was Involved
When my books—and then the fractal art—became known to seemingly everyone, my good eye was hardly ever praised. Instead, my merely being at IBM led to the perception that I was a lucky and passive beneficiary of an unfair competitive advantage.
In fact, I was not. Many other labs offered graphics off-the-shelf, but when I joined in 1958, IBM manufactured none, and to get outside equipment was harder than improvising. So on my way to achieving an unexpected status as a pioneer of computing—without ever touching the computer myself, but always giving instructions to programmers and assistants—I was forced to hustle ceaselessly.
Altogether, computer graphics took even longer than FORTRAN to become available. By the late 1960s, the most primitive helped me draw the first coastlines of artificial fractal islands. Our program simulated the whole relief. We could not visualize that “forgery,” but visualizing the coastline was possible. We were working on a grid of sixty-four by sixty-four pixels, and the first step consisted of leaving blank all the pixels that, together with their immediate neighbors, were either above or below sea level. The untouched points defined an approximate coastline. The output device was an ordinary typewriter, and the idea was to print the points on the coastline by superposing the letters M, W, and O (or something like that). I copied this printed output and then blackened the inside of the island using a felt marker.
(Illustration Credit 21.1)
The process was heroic because the “software” needed to type the M, W, and O was not properly documented. Also, the buffer memory was tiny: having printed sixty-four bytes, the program stopped—until the word “return” was typed by hand. Desperate, I begged my assistant to type it again and again, as long as it took to get an output. When I went home at the end of the day, he stayed on. The next morning, the desired output was waiting for me.
A subsequent early device was manufactured by a company called Calcomp. It consisted of a sheet of paper rotating around a cylinder and a pen that could be lifted or lowered onto that paper but could only move along the cylinder’s axis. A program combined the motions of the pen and the cylinder in an excruciatingly slow process, and the patterns it could draw were limited. We were pushing the machine well beyond its original specifications.
At long last, around 1970, graphic devices changed from mechanical to electronic. So a figure computed on the big mainframe could be delivered to a very shaky special purpose computer that allowed it to be examined on a laboratory cathode ray tube (CRT)—like an ordinary television screen. A special attachment made it possible to photograph the screen with a Polaroid camera. The first published pictures obtained in that way were the earliest fractal mountain reliefs.
(Illustration Credit 21.6)
A third graphics system, Lblgraph, came to life serendipitously in the early 1970s when IBM stopped an ill-starred foray into computer-based typesetting—and several colleagues and I changed it from black and white into sixty-four shades of gray.
(Illustration Credit 21.2)
To compensate for Yorktown’s geographic isolation—and to help the world know that we existed—a steady flow of visitors were invited to lecture, entertain, and educate. No striking picture was ever shown, and I was dying to go to color graphics, but my immediate management turned me down: IBM was not in that business, and purchasing competitors’ products was difficult.
One day in 1976, the grapevine reported that an outside supplier had installed our dream color graphics device for a colleague in the development group. On the spot, I called him and walked over for a chat. “Could we have access to this machine, and if so, when and how often?” “Of course. Gladly. But you must know that there is absolutely no software. I can pay a systems programmer, but to get one will take six months. Writing the software will take another six months. Come to see me a year from now.” “Well, well … we actually are in a bit of a hurry. Could we perhaps have your lab’s key code and stop over this weekend to get to know your gadget?” “Sure, why not.” So before leaving on Friday, he gave the code to my very close colleague, Richard Voss, who went immediately to work, likely taking no time off for sleep. The software was ready on Monday morning. One year of waiting had melted into one weekend!
(Illustration Credit 21.3)
So, why study coastlines? Initially, I picked them because nobody had a permanent interest that would interfere with their acceptance, but also because my father was a map nut. From him, I learned to read maps before I could read and write. One of the most striking features of fractals is that they enable us to imitate nature. After the first general idea of coastlines, I thought of constructing random coastlines from a simple formula, and then random landscapes. Without computer graphics, it would have been a herculean task.
Within a university, that color graphics device would have belonged to the NSF through the project that paid for it. Therefore, its use would have been severely constrained. But at Yorktown, all the tools belonged to IBM and were assigned to projects as needed. That funding method had the advantage, within limits and with proper justification, of giving everybody access to equipment—if assigned to close enough friends.
In joining IBM in 1958, I resumed—on a far larger scale—a lifestyle I had once known in the past. The small Paris laboratory of Philips was replaced by the huge IBM Research, and an undemanding graduate school was replaced by academic nomadism: a sequence of visiting professorships in distinct and very different fields and of “traversals” of seemingly incongruous areas of research. They appeared at first sight to clash badly—but they really didn’t clash at all. It soon emerged that I was working on the building blocks of my soon-to-be fractal geometry of nature.
22
At Harvard: Firebrand Newcomer to Finance Advances a Revolutionary Development, 1962–63
MY INVOLVEMENT WITH THE BEHAVIOR of financial prices—absolutely unplanned—became a constant of my scientific life. This revolutionary development went on to inspire many later works of mine, scattered around seemingly unrelated fields, and led me, in due time, to put forward a sharp distinction between two very different states of randomness: the “mild” and the “wild,” and a third state I call “slow.”
For several years—with IBM being focused on growth and the continual reorganization of its research division—I did little to be noticed. My first major piece of new work at IBM was embodied by a long publication, Research Note NC87, dated March 26, 1962. I was acutely aware that my findings would have devastating consequences for the accepted standard theory of speculation. I was in a big rush to finish it—but I had low priority for secretarial assistance. I could not wait f
or it to be typed professionally, so I proceeded to do it on my own little typewriter—with two fingers! That report’s abominable typing was disregarded, and reactions to its contents were—by academic standards—lightning fast and strong.
(Illustration Credit 22.1)
Then a letter arrived, inviting me to teach economics at Harvard. Letter in hand, I rushed to see my manager, Ralph Gomory. Very pleased, he sent me to his manager, Herman Goldstine (1913–2004), director of mathematical sciences at that time. I had met him in Princeton when he was John von Neumann’s second in command; because I had been relatively unimportant at IBM, this was—since Princeton—the first real contact we had.
“What brings you here today?” “I would like to be granted a one-year leave of absence to teach at a university.” “You know that this department strongly believes in supporting teaching. I’d be delighted to grant this request, and please remember that we cover the difference between the IBM and university salaries.” “By the way, you left me no time to mention that I’m invited to teach economics.” “Oh! I would have expected statistics or applied mathematics, but this is fine. More important, you haven’t named the university that is inviting you?” “Oh, I am very sorry. I should have started by saying so. The invitation is from Harvard, and their offer is higher than my current salary.” At this point in our conversation, he became extremely agitated and reached for a pillbox.
Harvard Called and IBM Noticed Me
If status within IBM had been measurable, mine would have instantly jumped from well under the radar to well above—where it stayed. This jump was far more important than the accompanying raise in salary from well below the norm to slightly above.
Leaves from IBM became part of an ongoing arrangement. That IBM granted academic leaves was a novelty, and outsiders watched without a clue as industrial labs came and went, and often questioned my position’s durability. I wondered as well, but reasoned that a huge and fast-growing computer manufacturer genuinely needed frontier research.
I long put out of mind the fact that IBM failed to provide the university guarantee of tenure. But not for a moment did I forget that to remain stable and vertical, a bicycle must move sufficiently fast. In pedantic words, I made a clear distinction between the static stability that tenure would have provided and a dynamic form of stability that was unexpected, makeshift, and constantly affected by shifts in both IBM and the outside world.
From the day I informed Herman Goldstine of my first “call” from Harvard, two things started to become increasingly clear. Invitations from elite outside institutions strengthened my position inside, and my attractiveness to outsiders could not be taken for granted. I became acutely sensitive to the question, “What have you done lately for our science?” My record as an innovator needed to be constantly enhanced by conspicuous fresh achievements. Each visit to a corner of academia contributed in ways that neither IBM nor academia’s other corners could possibly match. But to stoke my “innovation furnace” at IBM was best.
As a result, times at IBM and times in Cambridge and elsewhere came to alternate, an order slowly emerged, and distinct works were reshaped as aspects of a whole—a fractal geometry of roughness. The sequence that followed, of sociological low points and intellectual highs, could not have been predicted.
How I Came to Study Price Variation
Let me stop to tell how I became fascinated with price variation, a completely new topic for me. The background resides in some work I had done earlier on an ancient topic—the law of distribution of personal income, which had been discovered in the 1890s by Vilfredo Pareto (1848–1923). That law—and my work—intrigued a few economists, and I was invited to speak at a Harvard seminar directed by Hendrik S. Houthakker (1924–2008).
Upon entering “Hank’s” office, I got a surprise that made that day one of the most memorable of my life. A peculiar diagram on his blackboard seemed to me nearly identical to one I was about to draw in my lecture! How was it, I soon asked, that something I had just discovered about personal incomes was already on display? “I have no idea what you are talking about. This diagram concerns cotton prices.” He had been working with a student before I arrived, and the blackboard had not yet been erased.
Why should the way income or wealth is spread throughout society relate to the ups and downs of the price of cotton? Why should both cases exhibit the same pattern of concavity and convexity? Could this reveal a deeper connection between these two aspects of economics—some odd truth lurking behind the charts? By then, mainstream writers on finance had rediscovered the old theory that prices vary as if by the toss of a coin. They were looking for evidence, but reliable historical records were hard to come by. Cotton was an exception.
For more than a century, the New York Cotton Exchange kept exacting daily records of prices as the vital commodity moved from the plantations of the Old South to the dark mills of the industrial North. Virtually all interstate trading was centralized at one exchange. This should have been an economist’s dream, but to Houthakker and his student, it proved a nightmare. There were far too many big price jumps and falls. And the volatility kept shifting over time. Some years prices were stable, other years wild. “We’ve done all we can to make sense of these cotton prices. Everything changes, nothing is constant. This is a mess of the worst kind.” Nothing could make the data fit the existing statistical model, originally proposed in 1900, which assumed that each day’s price change was independent of the last and followed the mildly random pattern predicted by the bell curve.
In short order, we made a deal: he’d let me see what I could do. He handed me cardboard boxes of computer punch cards recording the data. “Good luck. If you can make any sense of them, please tell me what you find.”
Back at IBM, the computing center assigned a programmer to analyze those records—as it had for income distributions. How many big price jumps, how many small? The wait being long because I was low on the priority list, I took the train to Manhattan, where the National Bureau of Economic Research was then located. Its library included many dust-covered books filled with tables of financial data—a treasure before the time of computers. Later, I obtained records from the U.S. Department of Agriculture in Washington, D.C. Gathering every available piece of data, I built an encyclopedia of cotton prices, daily, weekly, monthly, and annually, over more than a century.
What the computer helped me find was extraordinary. Houthakker’s view was confirmed: the price changes from one day to the next, one week or month or year to the next, did not behave as the 1900 model assumed. The variance misbehaved. Each time I added a price change to the data set, my estimate of the variance changed. It never settled down to one simple number of volatility. Instead, it roamed erratically. That was surprising, considering that the quality of the data itself could not be challenged. Moreover, there were too many big price jumps to fit the bell curve.
Two Pictures of Price Variation
How do prices vary on the organized markets called bourses, stock exchanges, or commodity exchanges?
For centuries, such markets have thrived without the benefit of a systematic mathematical model. The first such model was put forward in 1900 by an outsider in French mathematics, Louis Bachelier (1870–1946). It came out astonishingly early—well ahead of its time—and was odd indeed. It became the standard financial model, and was the one Houthakker was using with cotton prices. The model was advanced financially, but not buttressed by any data whatsoever. Originally, it drew little attention, but over time two events revived it. On the theoretical side, Norbert Wiener rediscovered it around 1920 as a model of an important phenomenon called Brownian motion, which became more developed in the 1930s and ’40s. On the concrete side, the advent of the computer in the 1960s empowered the study of both the data and the theory.
Thanks to the computer, I was able to note the flaws in Bachelier’s model in a rough report I wrote in 1962, and put forward a counter theory that could be stated with no formula, which truly
fit my wild Keplerian dream. And it produced in 1963 my first paper on this work, “The Variation of Certain Speculative Prices,” which was to be frequently cited in the economics literature. The 1900 theory assumed that price jumps can be neglected—the mathematical concept being “prices vary continuously”—and that price changes follow the same rules in prosperity or depression. Much well-documented evidence contradicting this theory made frequent and large ad hoc “fixes” necessary. My 1962 counter theory allowed for discontinuities and was later extended in 1965 and 1973 to allow for alternating periods of prosperity and recession.
All price charts look alike. Sure, some go up, some down. But daily, monthly, annually, there is no big difference in their overall look. Strip off the dates and price markers and you cannot tell which is which. They are all equally wiggly. “Wiggly” is hardly a scientific term—and until I developed fractal geometry years later, there was no good way to quantify so vague a notion. But that is exactly what we can now see in the cotton data: a fractal pattern. Here, the fractal scaling up and down is not being done to a shape, such as the florets of a cauliflower. Rather, it is being applied to a different sort of pattern, the way prices vary. The very heart of finance is fractal. So it all comes full circle. It was no coincidence that Houthakker’s cotton chart looked like my income chart. The math was the same.
The Fractalist Page 22