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

14 JULY 2011 | VOL 475 | NATURE | 169 © 2011 Macmillan Publishers Limited. All rights reserved