Cold and calculating. A Dorid nudibranch (Tritoniella belli) in Antarctica. Photo by Norbert Wu/Minden/National Geographic
The world is full of mundane, meek, unconscious things embodying fiendishly complex mathematics. What can we learn from them?
What does it mean to know mathematics? Since maths is something we teach using textbooks that demand years of training to decipher, you might think the sine qua non is intelligence – usually ‘higher’ levels of whatever we imagine that to be. At the very least, you might assume that knowing mathematics requires an ability to work with symbols and signs. But here’s a conundrum suggesting that this line of reasoning might not be wholly adequate. Living in tropical coral reefs are species of sea slugs known as nudibranchs, adorned with flanges embodying hyperbolic geometry, an alternative to the Euclidean geometry that we learn about in school, and a form that, over hundreds of years, many great mathematical minds tried to prove impossible.
Sea slugs have at least the rudiments of brains; they generally possess a few thousand neurons, whose large size has made these animals a model organism for scientists studying basic neuronal functioning. This tiny number isn’t nearly enough to enable the slug to formulate any representation of abstract signs, let alone an ability to mentally manipulate them, and yet, somehow, a nudibranch materialises in the fibres of its very being a form that genius-level human mathematicians didn’t discover until the 19th century; and when they did, it nearly drove them mad. In this instance, complex brains were an impediment to understanding.
Nature’s love affair with hyperbolic geometry dates to at least the Silurian age, more than 400 million years ago, when sea floors of the early Earth were covered in vast coral reefs. Many species of corals, then and now, also have hyperbolic structures, which we immediately recognise by the frills and crenellations of their forms. Although corals are animals, they have only very simple nervous systems and can’t be said to have a brain. A head of coral is actually a colonial organism made up of thousands of individual polyps growing together; collectively, they grow a vascular system, a respiratory system and a crude gastrointestinal system through which all the individuals of the colony eat and breathe and share nutrients. Nothing like a brain exists, and yet the colony can organise itself into a mathematical surface disallowed by Euclid’s axiom about parallel lines. Strike two against ‘higher intelligence’.
Ask any fifth-grader what the angles of a triangle add up to, and she’ll say: ‘180 degrees’. That isn’t true on a hyperbolic surface. Ask our fifth-grader what’s the circumference of a circle and she’ll say: ‘2π times the radius’. That’s also not true on a hyperbolic surface. Most of the geometric rules we’re taught in school don’t apply to hyperbolic surfaces, which is why mathematicians such as Carl Friedrich Gauss were so disturbed when finally forced to confront the logical validity of these forms, and hence their mathematical existence. So worried was Gauss by what he was discovering about hyperbolic geometry that he didn’t publish his research on the subject: ‘I fear the howl of the Boetians if I make my work known,’ he confided to a friend in 1829. To their universal horror, other mathematicians soon converged on the same conclusion and the genie of non-Euclidean geometry was let loose.
But can we say that sea slugs and corals know hyperbolic geometry? I want to argue here that in some sense they do. Absent the apparatus of rationalisation and without the capacity to form mental representations, I’d like to postulate that these humble organisms are skilled geometers whose example has powerful resonances for what it means for us humans to know maths – and also profound implications for teaching this legendarily abstruse field.
I’m not the first person to have considered the mathematical capacities of non-sentient things. Towards the end of Richard Feynman’s life, the Nobel Prize-winning physicist is said to have become fascinated by the question of whether atoms are ‘thinking’. Feynman was drawn to this deliberation by considering what electrons do as they orbit the nucleus of an atom. In the earliest days of atomic science, atoms were conceived as little solar systems with the electrons orbiting in simple paths around their nuclei much as a planet revolves around its sun. Yet in the 1920s, it became evident that something much more mathematically complex was going on; in fact, as an electron buzzes around its nucleus, the shape it makes is like a diffused cloud. The simplest electron clouds are spherical, others have dumbbell and toroidal shapes. The form of each cloud is described by what’s called a Schrödinger equation, which gives you a map of where it’s possible for the electron to be in space…