Our concept of living in a universe with a knowable and predictable future has taken two stunning blows in the last century, first from quantum mechanics in the 1920s, which uncovered a number of quantities which don’t play well together and place limits on the degree to which we can measure the world around us, and then, with far less fanfare but with potentially even greater repercussions, from some odd mathematical results discovered by Mary Lucy Cartwright (1900-1998) and JE Littlewood while investigating the behavior of radar signal amplifiers during World War II. Their results, worked out in rigorous detail over the next decade, were the precursors of an entirely new approach to what science could and could not predict, a field today known as Chaos Mathematics.
For most of us, our knowledge of chaos mathematics more or less begins and ends with Jeff Godlblum’s speech while sprinkling droplets of water on Laura Dern’s hand in Jurassic Park – minute unobservable changes can add up in ways that make predicting macroscopic events impossible. We might even know the “butterfly effect” of Edward Lorenz’s thought experiment – “If a butterfly flaps its wings in Brazil, could it cause a tornado in Texas?” The situation that presented itself to Cartwright in 1938, however, was decidedly less poetic, and entered not on the wings of a lovely insect or through Goldblum’s dulcet tones, but via misbehaving radar equipment.
With World War II blooming in red ruin upon the horizon, England was busily employed attempting to get its radar technology to a consistently accurate standard. Airplanes, being made out of metal, are particularly good at reflecting radio waves that are sent in their direction. One could, by sending out a radio wave, and measuring the frequency change of the returning reflected wave, come to a conclusion about where an enemy plane is, and where it is headed. The problem was that reflected signals tended to be very weak, and needed amplification in order to be detected by operators. During the 1930s, however, operators began discovering that, when they required high degrees of signal enhancement, their amplifiers started going wacky, transforming the uniform input signals they were receiving into a cacophony of unpredictable, multi-frequency outputs that seemed to follow no pattern.
There were two ways to solve this issue, one was just to keep brute forcing the problem until a configuration was found that produced reliable results, but there was no telling how long that would take. The other was to send the problem off to mathematicians, and see if they could, by analyzing the mathematical properties of amplifier systems, come up with an explanation of the phenomenon and therefore potentially a targeted solution. In 1938, the problem fell in the laps of Cartwright and Littlewood, two function theorists who seemed an unlikely pair to tackle such an applied problem, but who turned out to be exquisitely well tailored for the job.
Mary Lucy Cartwright was born December 17, 1900 in the village of Aynho, in West Northamptonshire, to a family of distinguished lineage, including the poet John Donne and Royal Society curator (and planetarium inventor) John Theophilius Desaguiliers as ancestors. Her father was a rector, and the family had the means to provide young Cartwright with governesses until the age of eleven, when she began attending Leamington High School, where she stayed with relatives during the school week, returning home for weekends. She remained at Leamington from 1912 to 1915, then was a boarding student at Graveley Manor School for a year, before having the course of her life changed at Godolphin School in Salisbury, which she attended from 1916 to 1919.
At Godolphin, she was taken under the wing of a teacher who had taught herself the higher mathematics that were often omitted from a young woman’s education at even the finest of high schools. That teacher, a Miss Hancock, saw Cartwright’s raw talent and took the sixteen year old student under her wing, teaching her calculus, analytic geometry, and, most crucially for her future career, the theory of function convergence. In 1919, armed with a good background in pure mathematics, but hardly any in applied math, Cartwright began attending St. Hugh’s College at Oxford, where she quickly found her way into the classroom of one of the century’s great pure mathematicians, G.H. Hardy. She attended his late night evening classes, where students would discuss interesting mathematical topics from 8:45 to 11 pm. Here she truly caught the bug for function analysis that would lie at the center of so much of her work in the coming decades.
Graduating in 1923, Cartwright turned to teaching, first in Worcester then Buckinghamshire, but by 1927 she was back at Oxford as a graduate student, with Hardy as her advisor. Her work of the late 1920s centered on Abel summation, which is worth talking about because elements of this early work will reappear in her more famous later papers. Suppose you have a list of numbers that follows a pattern that you are able to write down. For example, the list 1, 1/2, 1/3, 1/4, 1/5,… can be written as following the pattern 1/n. The question becomes, if I want to add up all the terms of that pattern, onwards to infinity, will they sum to a finite number, or will the sum just keep getting bigger and bigger? Depending on what the pattern is, these questions can either be pretty easy to answer or very very tricky. Abel summation is a way of finding the sum of a pattern by representing that sum as a related integral, a mathematical object we have any number of techniques for evaluating.
What Cartwright was developing in this early work was a way of thinking that bridges the spaces between the behavior of integrable functions and the summability of the patterns related to them. She was to expand this type of thinking in the 1930s with her work on function theory and particularly with integral functions. She was interested in questions about the behavior of different types of functions restricted to a given region of space (often a circle of radius 1 centered at the origin known as the “unit circle”). Suppose, for example, you and a friend start at two different locations on the edge of that circle, and weave your way however you want towards the same point, and at every step along the way you evaluate an equation which takes as its input the coordinate of the location you happen to be standing on – what can you say about the outputs of those equations obtained by you and your friend as you both get closer and closer to the shared end point of your journeys?
Cartwright’s work of this era, building off earlier efforts by Finnish topologist Ernst Lindelof and her colleague J.E. Littlewood, focused on establishing boundaries on the behavior of functions defined on different spaces under different restrictions. If you have a box in your hand that displays different numbers as you walk through a space, which are arrived at through an equation located in the box, how do restrictions on the equation in the box place bounds on the numbers on the display, or determine how many times that number will come up zero? Cartwright was interested in finding increasingly precise and efficient ways of ensuring that that number on the box display won’t blow up to infinity no matter where you go in the space. She famously established conditions for the equation inside the box under which, if you saw that the displayed number was under a certain value for any integer valued input, you could then conclude that it would also be under that value for any real number valued input. So, if the box is behaving when you input 1, 2, 3, or 500, you know it will also behave if you plug in 1.8, or the square root of two, or 5 11/13, which brings answering questions about real number functions down to answering questions about behavior at the integers, which are generally far more pleasant to answer.
In 1930, Cartwright received her doctorate, and began her association at Girton College, which was to last for her entire career, serving first as a research fellow, then college lecturer, and by 1936 as Director of Studies for Mathematics. It was while in this position that the radar problem was dropped in her lap. Seemingly, nothing could be further from the field of expertise of an abstract function theorist than an applied question springing from electrical engineering, but in truth Cartwright’s background was perfect for cutting through the conundrum. By 1938, she had been grappling for a decade with problems about the conditions required for functions to behave reasonably on given spaces, and the radar problem could be viewed from precisely this perspective.
In 1927, a Dutch electrical engineer named Balthasar van der Pol had noticed while working with vacuum tubes that when they were driven at certain frequencies, irregularities could be heard in the output signal. He couldn’t know it yet, but those irregularities were the result of chaos effects which lay deep in the structure of the mathematical formula he created to describe the waves produced by his vacuum tubes. That formula was a differential equation, x’’-k(1-x^2)x’ + x = bk cos (Lt), where the right side of the equation describes the presence of external disturbing factors, and x(t) describes the position of the wave at any given moment in time, with x’ and x’’ being the first and second derivatives of that function.
Solving a differential equation involves finding an equation x(t) which meets the given criteria. So here, for example, Cartwright’s challenge was, given values of k, b, and ?, to find an equation such that, when you take the second derivative of it at any moment in time, and then subtract off k times the first derivative of it at that time multiplied by 1 minus its own value squared, and THEN add its own value, you end up with bk cos (?t) at that time. It’s a daunting task at the best of times, which Cartwright and Littlewood approached by investigating a more general form of the van der Pol equation, doing what they did best – establishing parameters for good behavior.
They considered the general differential equation x’’ + kf(x,x’) + g(x,k) = p1(t) + k p2(t) and wondered what would happen as you started tinkering with each part. What they found was that, as long as g(x,k) is relatively linear (i.e. the output is regularly proportional to the input), and as long as k is large, then the solution x(t) to the differential equation converges to a single result over time, but that when you start to move away from linear g’s and large k’s, convergence is by no means guaranteed, and in fact can produce numbers of different possible waves of uncertain stability.
Cartwright and Littlewood’s analysis of the radar problem took seven years to work out, by which time engineers had produced their own solution to the amplification problem using trial and error methods, but the resulting paper, 1945’s “On non-linear differential equations of the second order I” was a tour de force that the pair would expand on throughout the 1940s, and which played no little part in Cartwright’s election as a Fellow of the Royal Society in 1947, followed closely by her promotion to Mistress of Girton College in 1948, a position she held until 1968, shepherding the institution smoothly through an era of considerable social change.
When she retired from Girton in 1968, Cartwright spent the next three years visiting foreign universities before settling down in 1971 to her lodgings at 38 Sherlock Close in Cambridge where she spent the next two decades of her life, continuing her mathematical researches, producing her final paper, a return to the Van der Pol equation, in 1987. She had lived her days with an efficiency and emotional reserve which allowed her to accomplish much as both an administrator and researcher, but which also kept most colleagues and students at a somewhat professional distance. To the mathematical community, she was known and respected for her exquisite results on how functions behave near singularities, the prerequisites for uniform function behavior, and on how to employ conformal mappings to establish limits on the roots and asymptotic values of functions, but her pioneering role in detailing the mathematical behavior of a chaotic system was buried in such an intimidating wall of functional reasoning that it failed to capture the public imagination in the same way that the visually arresting sets of Mandelbrot or the accessible explanations of Lorenz did, causing this pioneer in the field of chaos mathematics to often be either omitted entirely from its history or relegated to a (sometimes very literal) footnote. Fortunately, that situation is beginning to change, and the road to Jeff Goldblum is increasingly placed less in the flutter of a butterfly’s wings and more in the squelch of a vacuum tube, straining to have its subversive deviations understood by a world not yet ready for them.
FURTHER READING: There are two very nice memoirs of Cartwright’s life that are relatively accessible, the first being WK Hayman’s Royal Society memoir upon her death, which is available online, and the second an appreciation by none other than Freeman Dyson in the pages of the wonderful collection Out of the Shadows: Contributions of Twentieth-Century Women to Physics (2006) edited by Nina Byers and Gary Williams. Of course, one would think that, given the importance of her analysis of the Van der Pol equation, she would be prominently featured in what most people consider the go-to popular account of chaos theory, James Gleick’s 1987 book Chaos: Making a New Science, but she appears in that volume as only a mention stuffed into an endnote, while Gleick focuses more on the work of Smale and Lorenz as the founders of chaos theory. Maybe this will change in future printings, but as of the 26th printing, it is still the case.