“Unlike most mathematical discoveries, however, no one was looking for a theory of groups or even a theory of symmetries when the concept was discovered. Quite the contrary; group theory appeared somewhat serendipitously, out of a millennia-long search for a solution to an algebraic equation.” – Mario Livio
Prolegomenon: The general perception of mathematics and mathematiciansis one of rigour, exactitude, conservatism. Mathematicians have a reputation for being a bit straight laced and predictable; there doesn’t really seem to be much room for the element of surprise or luck in their work. But really, as far as research mathematics goes, nothing could be further from the truth. In fact, mathematicians have firecrackers popping off in their heads. They are deeply creative thinkers. For in order to come up with original solutions to unsolved problems, they need to connect the unexpected. In order to invent algorithms that perform in ways not yet imaginable, they must explore the surprising. In this way, serendipity is as ubiquitous as it is omnipresent in mathematical research. It is the modus operandi for the best sort of mathematical insight. In the following we explore three notable moments of serendipity in mathematics through an allegory, just as sixteenth century Italian author Tramezzino explored the scientific method through a fable about the travels of three princes whom, by serendipity, were able to reconstruct the circumstances and location of a camel missing in the desert.
MATHEMATICIAN ONE: THE DOODLING OF STANISLAW ULAM
Prime numbers have been a source of fascination for mathematicians for millennia. A prime number is one that has only itself and 1 as a divider. For instance, the number 7 is a prime because the only numbers which divide it are 7 and 1. In contrast, 9 is not a prime, because in addition to itself and 1, it has 3 as a divisor. While we can identify many small prime numbers relatively easily, such as 3, 5, 7, 11, 13, 17 and so on, the further on up the number line we go the harder it gets. In essence, we have to check every number smaller than the number in question to make sure it isn’t a divisor. This is very slow for humans, but more importantly it is also rather slow for computers too, especially with very large numbers. “What is the 50th prime number?” one might ask. The only way we know how to answer this question is to produce a list of prime numbers and then take the 50th on that list. One of the major questions mathematicians thus ask is, “is there a good way to know where and how I might find more prime numbers?”
Throughout history, many interesting patterns have been noticed for the distribution of primes. One of the most serendipitous of these is a pattern discovered by Polish mathematician Stanislaw Ulam. As the story goes, Ulam was sitting through a rather monotonous and lengthy lecture and started doodling on the page to distract himself. However, this particular doodle was not an ordinary one; Ulam’s doodle began with arranging the counting numbers, 1, 2, 3 and so on in a spiral shape on the page forming a lattice (see above image). Once he got bored of doing this, Ulam decided to shade the prime numbers in the lattice, one after the other. After several dozen numbers, what Ulam found when he scanned the whole page astonished him! What jumped out of the page was a distinct visual pattern; clear diagonal, horizontal and vertical lines could be made out in the shadings. Primes appeared to concentrate around these lines, particularly the diagonal ones. The lattice and its resulting pattern is now known as Ulam’s spiral, and the visual image of primes rapidly became famous. His spiral also prompted researchers to consider similar graphical representations of primes and more serious mathematical research has gone into explaining this phenomenon. One wonders what would have happened had the lecture he was attending had not been so boring!
MATHEMATICIAN TWO: PHILIP GOOD’S LOST LABELS
“121, 118, 110, 34, 12, 22,” statistician Philip Good read out off six unmarked petri dishes before him and slumped back into his chair in disbelief. These numbers were the results of experimental research concerning the population of cells grown in the lab. After years in the safe confines of the mathematical office, Good had decided to branch out and merge his statistical skills directly with lab work. The latest task involved experimenting with aging cells that were grown in petri dishes; he treated half of the resulting samples with a regular nutrient solution, and the other half with designer solution which promised to produce ‘life extending’ properties. After weeks of careful maintenance and observation, six of the eight original dishes were viable and the results were assessed. However, the culmination of several years of planning and training was seemingly dashed when the lab assistant came sheepishly to deliver him some bad news: “I’ve lost the labels” the assistant mumbled. What? No labels? What a catastrophe! Without the labels, Good had no way of telling which petri dish had been treated with which solution, and the experiment was rendered seemingly useless. What made things worse is that the numbers seemed that they could support a favourable outcome for the ‘designer’ solution. If populations of 121, 118, and 110, could in fact be associated with the life-extending solution, Good would have made a huge discovery indeed. But how could this be established without the labels?
In desperation, Good transported himself out of the lab with its dishes and labels and back to his world of statistics and hypothesis testing. Back in this abstract realm of probabilities and likelihoods, Good developed a whole new line of statistical approaches and formalised decision-making techniques which allow him to claim with varying degrees of confidence which dish should have had which label. Inspired by these lost labels, Good made insights that have since paved the way for hypothesis testing throughout mathematics for various scenarios, not just biological ones. Without this moment of serendipity, albeit initially disastrous, Good might never have been motivated to consider experimental data and interpreting results in such a way at all, and therefore, never developed such important statistical tools for future generations of mathematicians. The accidental loss of labels thus prompted key discoveries. (The labels themselves were never found.)
All great discoveries include as an ingredient an element that is out of the discoverer’s control.
MATHEMATICIAN THREE: YVES MEYER FINDS THINGS HE THOUGHT SHOULDN’T EXIST
Office gossip of varying kinds can have a big impact, as French mathematician Yves Meyer found out. Prior to some banter at the office with colleagues, Meyer had never heard of wavelets, but they made him a household name in the field of signal processing. Indeed, wavelets are a relatively new branch of mathematics, having been developed over the last two decades to enhance digital technologies among other things. They are crucial in a wide range of practical applications including graphics, audio signals, and edge detection. The field has been advanced by researchers from many different specialities, including engineering, computer science, physics, pure and applied mathematics.
As the story goes, Meyer, who was a researcher at the École Polytechnique in Paris, heard about a recent advance in wavelet theory while standing in line for the photocopier. This coincidence generated much enthusiasm in him. He knew of analogous type structures but in a branch of pure mathematics called harmonic analysis. Inspired by this conversation, Meyer dedicated his researches to this new topic. In particular, he discovered a new type of wavelet quite by accident. Having set himself the task of proving that a wavelet with certain types of properties didn’t exist, he instead found a series of wavelets with exactly these properties. In this way, while looking to verify the absence of something, through this very search, he found the exact opposite to be the case. Meyer’s moment of serendipity lies in the fact that a search of any other kind may not have caused him to find these new series of wavelets, which heralded a breakthrough for the field.
These three episodes may seem remarkable, but there isn’t anything particularly special about the tales of these three mathematicians. Every research mathematician can readily offer their own stories of coincidence in the process of mathematical invention. All great discoveries include as an ingredient an element that is out of the discoverer’s control. Despite its logical foundation and pedantic predictability, mathematics arises through the synthetic function of reason. Mathematical insight is thus a delicate balance between chance and certainty. Indeed, a doodle, an oversight, or chasing after a false conviction are but three of the ways in which a stroke of serendipity performs its whimsical magic in the generation of new mathematical knowledge.
Daubechies, Ingrid. “Where do wavelets come from? A personal point of view” Retrieved from http://perso.ens-lyon.fr/paulo. goncalves/pub/WaveletsHistoryByDaubechies.pdf (2017-12-16).
Gardner, Martin. 1964. “Mathematical games: The remarkable lore of the prime number”, Scientific American, 210 (March), 120—128.
Good, Phillip. 2005. Permutation, Parametric, and Bootstrap Tests of Hypotheses. Springer series in Statistics, Boston: Springer (3rd edition).
Hodges, E. J. 1964. The Three Princes of Serendip. Atheneum, New York Livio, Mario. 2005. The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. Simon & Schuster.
Clemency is an associate professor at the school of Mathematics and Statistics at the University of Canterbury, Christchurch, New Zealand. She is a graduate from the Department of the History of Mathematics, Brown University, USA, which she completed as Fulbright scholar. Her consideration of the mathematical achievements of early cultures is carried out by the examination and analysis of primary source material in Sanskrit, Arabic, Greek, Latin, and Cuneiform. Her first book, Chasing Shadows-Mathematics, Astronomy, and the Early History of Eclipse Reckoning, focusing on the theoretical treatment of eclipse phenomena in the ancient world, was recently published by Johns Hopkins University Press. She is currently immersed in an international project on the history of mathematical astronomy in Sanskrit sources supported by a five-year Rutherford Discovery Fellowship awarded by the Royal Society of New Zealand. Her work involves travelling to India to locate Sanskrit manuscripts, reading the mathematics they include, and making their contents available to the scholarly world and beyond.
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Ulam Spiral number of dividers 100,000 first natural numbers, 2009. Author: Cortexd. Image generated in PHP with the GD Library.