In 1735, the Swiss mathematician Leonhard Euler solved a very important mathematical problem. It involved taking a walk around the Prussian city of Königsberg – now Kaliningrad in Russia – which was situated on the banks of the river Pregel and included two large islands. These four pieces of land were connected by seven bridges. Euler, considered widely to be the greatest mathematician of his time, had some time earlier asked if anyone could take a walk over each of the seven bridges no more than once and visit all four parts of the city. On August 26, 1735, he answered himself during a presentation at the St. Petersburg Academy. Such a walk was impossible.
Because of the way the parts of the city were arranged and connected by the bridges, Euler first simplified the map into a drawing: instead of a planner’s blueprint, he represented each part as a node and the bridges as lines connecting the nodes. The resulting network has since been called a graph, and that moment was the birth of graph theory. Next, Euler proceeded to show that, if he had to traverse the seven bridges of Königsberg just once and visit all four parts of the city, there would have to be either no or two parts of the city that were connected by an odd number of bridges. If this condition was met, then Euler would be able to take his particular walk, since called the Eulerian path. However, all four parts were connected by an odd number of bridges (one by five and three by three each). No Eulerian path was possible.
Apart from seeding graph theory, Euler also helped popularise the field of study called topology. The graph that he had generated of Königsberg before solving his problem could be stretched, compressed or twisted around – but its fundamental arrangement of nodes and connections wouldn’t change. The study of those properties that can’t be altered by such continuous transformations is called topology. For example, any other city could be topologically similar to Königsberg if it wanted to: all it would have to have is the same arrangement of lands and bridges, but of whatever shape and size.
The Nobel Prizes for physics and chemistry this year recognise achievements in topology – the former more than the latter. The physics prize was awarded to David Thouless, Michael Kosterlitz and Duncan Haldane for their theories of the topological states of matter. Like with the examples above: a topological state of matter is a state of matter distinguished by its topological properties over anything else. Of the three, Thouless’s contributions were especially significant because they introduced, and laid the foundation for, the participation of topological ideas in characterising the states of matter. His work provided his peers with new ways to control matter’s behaviour at very, very low temperatures using a leash of not-so-new mathematical techniques.
Similarity to metamaterials
It’s never a good idea to harp on the applications of such research as a means to popularising it if only because the practice often leads to misguided assumptions about why research is conducted in the first place. If everything had been application-oriented from the get-go, then on the bright side we might’ve been more efficient at engineering solutions today. But the crazy cost of this is that we would’ve precluded accidents, deprived ourselves of the opportunities to make unexpected discoveries, learnt new things and disabused ourselves of the notion that, to begin with, we knew what we wanted. We seldom do. And this practice is especially bad when the research itself might have been conducted purely as a matter of curiosity (as we’ll see with the chemistry laureates). Curiosity must be allowed to hold on to its intrinsic value.
However, this said, there is no better example to illustrate the significance of Thouless’s, Kosterlitz’s and Haldane’s work than to point at a wonderful product of it. On August 29, Chinese researchers reported through a peer-reviewed paper of a new kind of material called an acoustic topological insulator. A topological insulator is a kind of material that takes advantage of its topological properties to conduct electricity only on its surface; its innards remain stubbornly resistant to the flow of current. Scientists are now in pursuit of such materials that are also superconductors on the surface. An acoustic topological insulator is not quite similar, however: it’s a material that the Chinese researchers claim can almost completely eliminate backscattering.
When a wave – say, of light – is sent through a medium and parts of the medium reflect the wave back towards the source, that’s backscattering. It’s a common thing that happens whenever we make use of waves travelling through a medium. And when the waves are those of sound, then backscattering represents a loss in transmission because not all the sound waves sent into the medium emerge from the other end. Some are scattered back.
Instead of looking at all this from a “what problem could this solve” PoV, it might be more entertaining to think of topological materials in terms of their circumstantial proximity to metamaterials. At the outset, metamaterials are solution-oriented almost by definition: they’re materials that have been engineered to possess properties that can’t be naturally found. At the same time, their behaviour – for example, a prism that inverts light the other way or an invisibility cloak – is curious because all metamaterials are composed of materials whose behaviour is at all times natural. The meta-prism is composed of “an insulator material that has been embedded with a crisscrossing of metallic wires, like those made of copper, through which an electric current is passed.”
It’s the synecdoche that fails. When these well-understood materials come together the way they do, the collective turns strange by deliberately affecting the path of light through them. They’re cis-fantastic, their weirdness arising from the fact that they maintain some proximity to normal materials by possessing a variety of properties we’re able to intuit – but then possessing that one property that throws our engagement with it off kilter. In another sense, they also demonstrate what Thouless did through his adoption of topology: that though it was okay to try to understand individual components (like atoms or metallic wires) using the traditional rules of physics, figuring their behaviour when they came together wouldn’t be possible by just combining those rules. Instead, there would be something great, more wholesome, at play, especially when operating at the very low temperatures where quantum physics rules.
Recourse in mathematics
This is true also of topological insulators, whether superconducting or acoustic. Their weirdness is the weirdness of the quantum world but amplified to become perceptible at macroscopic levels. Though quantum mechanical phenomena are in fact natural, many of them strain the imagination, let alone intuition. And when they’re magnified to become perceptible by human senses, there’s a cognitive clash between how the material behaves and how we expect it to behave. It feels unexpected and unnatural because there’s something more than a simple combination of components at work. Quantum mechanical effects interfere with our cognition in ways similar to that of the anti-synecdochic effects of metamaterials. The interference is a break from familiarity and such familiarity is often the blissful ignorance of advanced math.
But then again, isn’t it also beautiful that we have been able to build such materials? In many realms of physics, especially in particle physics, physical explanations don’t exist for some phenomena. Instead, the scientist takes recourse through mathematical ideas to understand the ideas at hand. A popular example is higher dimensions: though we may never be able to physically access or even perfectly imagine five- or six-dimensional spaces, mathematicians have been able to flatten them onto symbols on a sheet of paper. As Evelyn Lamb described it in Scientific American using metaphors from sewing, a three-dimensional space can be described by three coordinates (length, breadth and height), so simply think of the sixth dimension as space that can be described using six coordinates.
Another example concerns the particles called neutrinos. When the Nobel Prize for physics in 2015 was awarded to Takaaki Kajita and Arthur McDonald for discovering that neutrinos can transform from one type to another while in motion, there was a scramble among many of my colleagues to understand why this happened – until we all gave up when we came up against a wall of mathematics. It is as Daniel Sarewitz wrote in 2012: “Most people, including most scientists, can acquire knowledge of the [Higgs boson, discovered earlier in 2012] only through the metaphors and analogies that physicists and science writers use to try to explain phenomena that can only truly be characterized mathematically.”
But while those who can’t understand the math have to repose their faith in metaphors, even those who did understand the math got it wrong until the 1980s, when better metaphors became available. There were those who believed that at very low temperatures, the atoms or molecules of a material wouldn’t possess enough thermal energy to undergo phase transitions, so the phenomena were ruled out. Thouless wasn’t satisfied by this explanation. He teamed up with Kosterlitz in 1973 to propose a landmark experiment in which a two dimensional material, like a sheet of electrons, was shown to undergo phase transitions not by heating up or cooling down but by the display of patterns of vortices in its arrangement. These topological changes were shown by Thouless to be proof of a new kind of phase transitions. In effect, physicists had to wait until David Thouless showed up to metaphorise quantum mechanics through the ideas of topology.
Beauty isn’t just elegance and naturalness but also the manifestation of inexplicability and predictability at once – an act of stumbling upon something that fits so snugly but you can’t really tell why.
And the show goes on
Sometimes, it so happens that you might be familiar with the ‘why’ but couldn’t quite be bothered with the ‘what for’. This is as with the winners of the Nobel Prize for chemistry this year. Jean-Pierre Sauvage, Fraser Stoddart and Bernard Feringa won for building nanomechanical systems: elevators, motors, cars composed of a few molecules each, akin to assembling the world’s smallest LEGO set. What for? Fully realised applications for these molecular machines are yet to emerge; they could be anything. For example, earlier this year, researchers from the Indian Institute of Science in Bengaluru built a microscopic Stirling engine and came away with insights into how a human body’s biological motors are so much more efficient than the motors humans build. Just like this, in their quest to build molecular chains, wheel-and-axle assemblages, gears and motors, Sauvage, Stoddart and Feringa threw light on how we could play around with one of the ‘materials’ that chemistry itself was made of.
Just as well, the trio’s attempt to build a molecular car, for example, was not exactly outcome-oriented. In other words, they didn’t simply want to build a something that could move across a surface. They specifically wanted a car (or parts of a car over the years), in that the product of their efforts would be built like a conventional car: mobilised by four wheels connected across axles and with a chassis. And in seeking out this resemblance, they had to create molecules that reproduced the topology of their larger counterparts. Stoddart constructed molecules linked like chains and built a ‘machine’ in which a molecule could move between two other molecules like a shuttle. Sauvage pioneered the construction of molecular muscles, i.e. groupings that could stretch and contract under continuous transformations just like human muscles could, with the application of electrical or thermodynamic stimuli. Feringa built a molecular motor that could spin only in one direction. If their collective outcome had been to build a molecular device that would move across a surface, they could’ve built it to look like anything, but in attempting to recreate a conventional car, they’ve enhanced our topological awareness of the participating molecules and how they can be created in the lab.
Why did they want to do this? Stoddart wrote in 2005 that he did simply because he was motivated by his admiration of the mechanical properties of chemical bonds. He finished, “What will they be good for? Something for sure, and we still have the excitement of finding out what that something might be. And so the story goes on…” When Sauvage was awarded the prize, Gérard Férey, an emeritus professor at the University of Versailles, San Quentin, remarked that Sauvage was a “creative” and “great designer of molecules”. According to Feringa, just making molecules that didn’t exist before is excitement enough for him. To be sure, these are Nobel laureates fiddling with trefoil knots and Borromean rings thousands of times smaller than the width of a human hair, in many circumstances quite unaware of what they could be good for.
Using the techniques of the physics and chemistry laureates, it seems for sure that the engineering of the future can be very fun and very fruitful, too. But at the same time, who’s to say that its practice won’t reveal yet more realms of matter that will force physicists to introduce new mathematical metaphors to explain them, and so uncover even more room for research, design and manipulation – just the way Kosterlitz and Thouless did in their experiment? In principle, this could be a never ending journey to the bottom, and no one knows what more weirdness we’ll find there. The only thing that’s certain is that the pursuit of these possibilities is enriched when not shackled by the need for findings to be immediately useful.