The Unplanned Impact of Mathematics

COMMENT

The unplanned impact of mathematics Peter Rowlett introduces seven little-known tales illustrating that theoretical work may lead to practical applications, but it can’t be forced and it can take centuries.

s a child, I read a joke about some- it contains a mistake. If it was true for Archi- without apparent direction or purpose is one who invented the electric plug medes, then it is true today. fundamental to the discipline. Applicability and had to wait for the invention The mathematician develops topics that is not the reason we work, and plenty that is Aof a socket to put it in. Who would invent no one else can see any point in pursuing, or not applicable contributes to the beauty and something so useful without knowing what pushes ideas far into the abstract, well beyond magnificence of our subject. purpose it would serve? Mathematics often where others would stop. Chatting with a col- There has been pressure in recent years displays this astonishing quality. Trying league over tea about a set of problems that for researchers to predict the impact of to solve real-world problems, researchers ask for the minimum number of stationary their work before it is undertaken. Alan often discover that the tools they need were guards needed to keep under observation Thorpe, then chair of Research Councils PARKINS DAVID BY ILLUSTRATIONS developed years, decades or even centuries every point in an art gallery, I outlined the UK, was quoted by Times Higher Educa- earlier by mathematicians with no prospect basic mathematics, noting that it only works tion (22 October 2009) as saying: “We have of, or care for, applicability. And the toolbox on a two-dimensional floor plan and breaks to demonstrate to the taxpayer that this is is vast, because, once a mathematical result down in three-dimensional situations, such an investment, and we do want research- is proven to the satisfaction of the disci- as when the art gallery contains a mezzanine. ers to think about what the impact of their pline, it doesn’t need to be re-evaluated in “Ah,” he said, “but if we move to 5D we can work will be.” The US National Science the light of new evidence or refuted, unless adapt …” This extension and abstraction Foundation is similarly focused on broader

166 | NATURE | VOL 475 | 14 JULY 2011 © 2011 Macmillan Publishers Limited. All rights reserved COMMENT impacts of research proposals (see Nature solving problems in geometry, mechanics 465, 416–418; 2010). However, predicting and optics. After his death the torch was GRAHAM HOARE impact is extremely problematic. The latest carried by Peter Guthrie Tait (1831–1901), International Review of Mathematical Sci- professor of natural philosophy at the Uni- From geometry ences (Engineering and Physical Sciences versity of Edinburgh. William Thomson Research Council; 2010), an independent (Lord Kelvin) wrote of Tait: “We have had to the Big Bang assessment of the quality and impact of a thirty-eight-year war over quaternions.” UK research, warned that even the most Thomson agreed with Tait that they would Correspondence editor, Mathematics theoretical mathematical ideas “can be use quaternions in their important joint Today useful or enlightening in unexpected ways, book the Treatise on Natural Philosophy sometimes several decades after their (1867) wherever they were useful. How- In 1907, Albert Einstein’s formulation of appearance”. ever, their complete absence from the final the equivalence principle was a key step There is no way to guarantee in advance manuscript shows that Thomson was not in the development of the general theory what pure mathematics will later find persuaded of their value. of relativity. His idea, that the effects of application. We can only let the process By the close of the nineteenth century, acceleration are indistinguishable from of curiosity and abstraction take place, let vector calculus had eclipsed quaternions, the effects of a uniform gravitational field, mathematicians obsessively take results to and mathematicians in the twentieth cen- depends on the equivalence between gravi- their logical extremes, leaving relevance far tury generally followed Kelvin rather than tational mass and inertial mass. Einstein’s behind, and wait to see which topics turn Tait, regarding quaternions as a beautiful, essential insight was that gravity mani- out to be extremely useful. If not, when the but sadly impractical, historical footnote. fests itself in the form of space-time cur- challenges of the future arrive, we won’t So it was a surprise when a colleague vature; gravity is no longer regarded as a have the right piece of seemingly pointless who teaches computer-games development force. How matter curves the surrounding mathematics to hand. asked which mathematics module students space-time is expressed by Einstein’s field To illustrate this, I asked members of the should take to learn about quaternions. It equations. He published his general theory British Society for the History of Mathemat- turns out that they are particularly valuable in 1915; its origins can be traced back to ics (including myself) for unsung stories for calculations involving three-dimen- the middle of the previous century. of the unplanned impact of mathematics sional rotations, where they have various In his brilliant Habilitation lecture of (beyond the use of number theory in mod- advantages over matrix methods. This 1854, Bernhard Riemann introduced the ern cryptography, or that the mathematics makes them indispensable in robotics and principal ideas of modern differential geom- to operate a computer existed when one was computer vision, and in ever-faster graph- etry — n-dimensional spaces, metrics and built, or that imaginary numbers became ics programming. curvature, and the way in which curvature essential to the complex calculations that fly Tait would no doubt be happy to have controls the geometric properties of space aeroplanes). Here follow seven; for more, see finally won his ‘war’ with Kelvin. And Ham- — by inventing the concept of a manifold. www.bshm.org. Peter Rowlett ilton’s expectation that his discovery would Manifolds are essentially generalizations be of great benefit has been realized, after 150 of shapes, such as the surface of a sphere years, in gaming, an industry estimated to be or a torus, on which one can do calculus. MARK MCCARTNEY & worth more than US$100 billion worldwide. Riemann went far beyond the conceptual frameworks of Euclidean and non-Euclidean TONY MANN geometry. He foresaw that his manifolds could be models of the physical world. From quaternions The tools developed to apply Riemannian geometry to physics were initially the work to Lara Croft of Gregario Ricci-Curbastro, beginning in 1892 and extended with his student Tullio University of Ulster, Newtownabbey, Levi-Civita. In 1912, Einstein enlisted the UK; University of Greenwich, London help of his friend, the mathematician Marcel Grossman, to use this ‘tensor calculus’ Famously, the idea of quaternions came to to articulate his deep physical insights in the Irish mathematician William Rowan mathematical form. He employed Riemann Hamilton on 16 October 1843 as he was manifolds in four dimensions: three for walking over Brougham Bridge, Dublin. space and one for time (space-time). He marked the moment by carving the It was the custom at the time to assume equations into the stonework of the bridge. that the Universe is static. But Einstein soon Hamilton had been seeking a way to extend found that his field equations when applied the complex-number system into three to the whole Universe did not have any static dimensions: his insight on the bridge was solutions. In 1917, to make a static Universe that it was necessary instead to move to four possible, Einstein added the cosmological dimensions to obtain a consistent number constant to his original field equations. Rea- system. Whereas complex numbers take the sons for believing in an explosive origin to form a + ib, where a and b are real numbers the Universe, the Big Bang, were put forward and i is the square root of −1, quaternions by Aleksander Friedmann in his 1922 study have the form a + bi + cj + dk, where the rules of Einstein’s field equations in a cosmological are i2 = j2 = k2 = ijk = −1. context. Grudgingly accepting the irrefutable Hamilton spent the rest of his life promot- evidence of the expansion of the Universe, ing the use of quaternions, as mathematics Einstein deleted the constant in 1931, refer- both elegant in its own right and useful for ring to it as “the biggest blunder” of his life.

These proofs are simpler than the result for produce a game in which the expected out- three dimensions, and relate to two incred- come is positive. The term ‘Parrondo effect’ ibly dense packings of spheres, called the is now used to refer to an outcome of two E8 lattice in 8-dimensions and the Leech combined events being very different from lattice in 24 dimensions. the outcomes of the individual events. This is all quite magical, but is it A number of applications of the Parrondo useful? In the 1960s an engineer called effect are now being investigated in which Gordon Lang believed so. Lang was chaotic dynamics can combine to yield non- designing the systems for modems and chaotic behaviour. For example, the effect was busy harvesting all the can be used to model the population dynam- mathematics he could find. ics in outbreaks of viral diseases and offers He needed to send a prospects of reducing the risks of share-price signal over a noisy chan- volatility. Plus it plays a leading part in the nel, such as a phone plot of Richard Armstrong’s 2006 novel, God line. The natural way is Doesn’t Shoot Craps: A Divine Comedy. to choose a collection of tones for signals. But the sound received may not be PETER ROWLETT the same as the one sent. To solve this, he described the sounds by a list of num- From gamblers EDMUND HARRISS bers. It was then simple to find which of the signals that might have been sent was clos- to actuaries From oranges est to the signal received. The signals can then be considered as spheres, with wiggle University of Birmingham, UK to modems room for noise. To maximize the information that can be sent, these ‘spheres’ must be In the sixteenth century, Girolamo Cardano University of Arkansas, Fayetteville packed as tightly as possible. was a mathematician and a compulsive In the 1970s, Lang developed a modem gambler. Tragically for him, he squandered In 1998, mathematics was suddenly in the with 8-dimensional signals, using E8 pack- most of the money he inherited and earned. news. Thomas Hales of the University of ing. This helped to open up the Internet, as Fortunately for modern actuarial science, he Pittsburgh, Pennsylvania, had proved the data could be sent over the phone, instead of wrote in the mid-1500s what is considered Kepler conjecture, showing that the way relying on specifically designed cables. Not to be the first work in modern probability greengrocers stack oranges is the most effi- everyone was thrilled. Donald Coxeter, who theory, Liber de ludo aleae, finally published cient way to pack spheres. A problem that had helped Lang understand the mathemat- in a collection in 1663. had been open since 1611 was finally solved! ics, said he was “appalled that his beautiful Around a century after the creation of this On the television a greengrocer said: “I think theories had been sullied in this way”. theory, another gambler, Chevalier de Méré, that it’s a waste of time and taxpayers’ money.” had a dilemma. He had been offering a game I have been mentally arguing with that green- in which he bet he could throw a six in four grocer ever since: today the mathematics of JUAN PARRONDO & rolls of a die, and had done well out of it. He sphere packing enables modern communica- varied the game in a way that seemed sensi- tion, being at the heart of the study of channel NOEL-ANN BRADSHAW ble, betting he could throw a double six with coding and error-correction codes. two dice in 24 rolls. He had calculated the In 1611, Johannes Kepler suggested that From paradox chances of winning in both games as equiva- the greengrocer’s stacking was the most lent, but found he lost money in the long run efficient, but he was not able to give a proof. to pandemics playing the second game. Confused, he asked It turned out to be a very difficult problem. his friend Blaise Pascal for an explanation. Even the simpler question of the best way University of Madrid; University of Pascal wrote to Pierre de Fermat in 1654. The to pack circles was only proved in 1940 by Greenwich, London ensuing correspondence laid the foundations László Fejes Tóth. Also in the seventeenth for probability theory, and when Christiaan century, Isaac Newton and David Gregory In 1992, two physicists proposed a sim- Huygens learned of the results he wrote the argued over the kissing problem: how many ple device to turn thermal fluctuations at first published work on probability, De Ratio- spheres can touch a given sphere with no the molecular level into directed motion: ciniis in Ludo Aleae (published 1657). overlaps? In two dimensions it is easy to a ‘Brownian ratchet’. It consists of a particle In the late seventeenth century, Jakob prove that the answer is 6. Newton thought in a flashing asymmetric field. Switching the Bernoulli recognized that probability theory that 12 was the maximum in 3 dimensions. field on and off induces the directed motion, could be applied much more widely than to It is, but only in 1953 did Kurt Schütte and explained Armand Ajdari of the School of games of chance. He wrote Ars Conjectandi Bartel van der Waerden give a proof. Industrial Physics and Chemistry in Paris and (published, after his death, in 1713), which The kissing number in 4 dimensions was Jacques Prost of the Curie Institute in Paris. consolidated and extended the probability proved to be 24 by Oleg Musin in 2003. In Parrondo’s paradox, discovered in 1996 by work by Cardano, Fermat, Pascal and Huy- 5 dimensions we can say only that it lies one of us (J.P.), captures the essence of this gens. Bernoulli built on Cardano’s discovery between 40 and 44. Yet we do know that the phenomenon mathematically, translating it that with sufficient rolls of a fair, six-sided answer in 8 dimensions is 240, proved back into a simpler and broader language: gam- die we can expect each outcome to appear in 1979 by Andrew Odlyzko of the University bling games. In the paradox, a gambler alter- around one-sixth of the time, but that if we of Minnesota, Minneapolis, and Neil Sloane. nates between two games, both of which lead roll one die six times we shouldn’t expect to The same paper had an even stranger result: to an expected loss in the long term. Surpris- see each outcome precisely once. Bernoulli the answer in 24 dimensions is 196,560. ingly, by switching between them, one can gave a proof of the law of large numbers,

168 | NATURE | VOL 475 | 14 JULY 2011 © 2011 Macmillan Publishers Limited. All rights reserved COMMENT which says that the larger a sample, the more use it to understand how galaxies form. useful grew rapidly to include acoustics, closely the sample characteristics match Mobile-phone companies use topology to optics and electric circuits. Nowadays, those of the parent population. identify the holes in network coverage; the Fourier methods underpin large parts of Insurance companies had been limiting phones themselves use topology to analyse science and engineering and many modern the number of policies they sold. As poli- the photos they take. computational techniques. cies are based on probabilities, each policy It is precisely because topology is free of However, the mathematics of the early sold seemed to incur an additional risk, the distance measurements that it is so power- nineteenth century was inadequate for the cumulative effect of which, it was feared, ful. The same theorems apply to any knotted development of Fourier’s ideas, and the reso- could ruin a company. Beginning in the DNA, regardless of how long it is or what ani- lution of the numerous problems that arose eighteenth century, companies began their mal it comes from. We don’t need different challenged many of the great minds of the current practice of selling as many policies as brain scanners for people with different-sized time. This in turn led to new mathematics. possible, because, as Bernoulli’s law of large brains. When Global Positioning System data For example, in the 1830s, Gustav Lejeune numbers showed, the bigger the volume, about mobile phones are unreliable, topol- Dirichlet gave the first clear and useful defi- the more likely their predictions are to be ogy can still guarantee that those phones nition of a function, and Bernhard Riemann accurate. will receive a signal. Quantum computing in the 1850s and Henri Lebesgue in the won’t work unless we can build a robust 1900s created rigorous theories of integra- system impervious to noise, so braids are tion. What it means for an infinite series to JULIA COLLINS perfect for storing information because they converge turned out to be a particularly slip- don’t change if you wiggle them. Where will pery animal, but this was gradually tamed From bridges topology turn up next? by theorists such as Augustin-Louis Cauchy and Karl Weierstrass, working in the 1820s to DNA and 1850s, respectively. In the 1870s, Georg Cantor’s first steps towards an abstract University of Edinburgh, UK theory of sets came about through analys- ing how two functions with the same Fourier When Leonhard Euler proved to the people series could differ. of Königsberg in 1735 that they could not The crowning achievement of this math- traverse all of their seven bridges in one ematical trajectory, formulated in the first trip, he invented a new kind of mathemat- decade of the twentieth century, is the concept ics: one in which distances didn’t matter. of a Hilbert space. Named after the German His solution relied only on knowing the mathematician David Hilbert, this is a set of relative arrangements of the bridges, not elements that can be added and multiplied on how long they were or how big the land according to a precise set of rules, with special masses were. In 1847, Johann Benedict properties that allow many of the tricky ques- Listing finally coined the term ‘topology’ tions posed by Fourier series to be answered. to describe this new field, and for the next Here the power of mathematics lies in the 150 years or so, mathematicians worked to level of abstraction and we seem to have left understand the implications of its axioms. the real world behind. For most of that time, topology was Then in the 1920s, Hermann Weyl, Paul pursued as an intellectual challenge, with Dirac and John von Neumann recognized no expectation of it being useful. After all, that this concept was the bedrock of quan- in real life, shape and measurement are tum mechanics, since the possible states of a important: a doughnut is not the same as quantum system turn out to be elements of a coffee cup. Who would ever care about just such a Hilbert space. Arguably, quantum 5-dimensional holes in abstract 11-dimen- mechanics is the most successful scientific sional spaces, or whether surfaces had one theory of all time. Without it, much of our or two sides? Even practical-sounding parts modern technology — lasers, computers, of topology such as knot theory, which had CHRIS LINTON flat-screen televisions, nuclear power — its origins in attempts to understand the would not exist. ■ structure of atoms, were thought to be use- From strings to less for most of the nineteenth and twentieth centuries. nuclear power CORRECTIONS Suddenly, in the 1990s, applications of In the Comment article ‘Buried by bad topology started to appear. Slowly at first, Loughborough University, UK decisions’ (Nature 474, 275–277), the but gaining momentum until now it seems statement “we will save lives by pushing a as if there are few areas in which topology Series of sine and cosine functions were trolley into a person but not a person into is not used. Biologists learn knot theory used by Leonard Euler and others in the a trolley” refers to an incorrect reference. to understand DNA. Computer scientists eighteenth century to solve problems, The correct one is J. D. Greene et al. are using braids — intertwined strands of notably in the study of vibrating strings Science 293, 2105–2108 (2001). material running in the same direction — to and in celestial mechanics. But it was build quantum computers, while colleagues Joseph Fourier, at the beginning of the The Comment article ‘Crowd control in down the corridor use the same theory to nineteenth century, who recognized the Rwanda’ (Nature 475, 572–573) should get robots moving. Engineers use one-sided great practical utility of these series in have stated that family-planning aid Möbius strips to make more efficient con- heat conduction and began to develop a dropped from 30% to 12% of overall veyer belts. Doctors depend on homology general theory. Thereafter, the list of areas health aid, not overall aid. theory to do brain scans, and cosmologists in which Fourier series were found to be

CORRECTION This article originally credited Andrew Odlyzko with publishing results on kissing numbers in 8 and 24 dimensions. It has now been corrected to reflect that this work was done jointly with Neil Sloane.