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. 14 JULY 2011 | VOL 475 | NATURE | 167 © 2011 Macmillan Publishers Limited. All rights reserved COMMENT 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.