The Mathematical Code Hidden In Nature

The living world is a universe of shapes and patterns—beautiful, complex, and sometimes strange. Beneath all of them lies a mystery: How does so much variety arise from the same simple ingredients: cells and their chemical instructions? There is one elegant idea that describes many of biology’s varied patterns, from spots to stripes and everything in between. It’s a code written not in the language of DNA, but in mathematics. Can simple equations really explain something as messy and unpredictable as the living world? How accurately can mathematics predict reality? Could there be one universal code that explains all of this?

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Hello, everyone! What color is a zebra? Is it black with white stripes or white with black stripes? This is not a trick question. The answer is that zebras are black with white stripes, and we know this because some zebras are born without their stripes. This raises the question: Why do zebras have stripes to begin with? A biologist might say that the stripes aid in camouflage from predators, but that would be incorrect. The stripes’ actual purpose is most likely to confuse biting flies.

But that answer only tells us what the stripes do, not where they come from or why patterns like this are even possible. Our best answer to those questions doesn’t come from a biologist at all. In 1952, mathematician Alan Turing published a set of surprisingly simple mathematical rules that can explain many of the patterns we see in nature, ranging from stripes to spots to labyrinth-like waves and even geometric mosaics. These are now known as “Turing patterns.”

Most people know Alan Turing as a famous wartime codebreaker and the father of modern computing. However, many of the problems that fascinated him throughout his life were about biology. But why would a mathematician be interested in biology in the first place?

That’s a great question! I’m Dr. Natasha Ellison from the University of Sheffield in the UK. Many mathematicians are drawn to biology because it’s so complicated, and there’s so much we don’t know about it. Consider a living system, like a human being—there are countless processes happening simultaneously, and we don’t know everything about them. The movements of animals, population trends, evolutionary relationships, interactions between genes, and how diseases spread are all biological problems where mathematical models can help describe and predict what we observe in real life. Mathematical biology is also useful for describing things we can’t see.

When people ask why we should care about math in biology, the answer is that there are aspects of biology that we can’t observe directly. We can’t follow every animal in the wild or measure every gene and chemical in a living organism at every moment. Mathematical models can help make sense of these unobservable phenomena. One of the most challenging things to observe in biology is how living things grow and acquire their shapes. Alan Turing referred to this as “morphogenesis,” the “generation of form.”

In 1952, Turing published a paper titled “The Chemical Basis of Morphogenesis,” which contained a series of equations describing how complex shapes can arise spontaneously from simple initial conditions. According to Turing’s model, all it takes to form these patterns is two chemicals that spread out like gas atoms filling a box and react with one another. Turing called these chemicals “morphogens.” However, there was one crucial difference: these chemicals spread out at different rates.

The way we create a Turing pattern involves equations called reaction-diffusion equations, which describe how two or more chemicals move around and react with each other. Diffusion is the process of spreading out. Imagine a dish with two chemicals in it; they both spread out and react with each other. This is what reaction-diffusion equations describe.

Turing’s first insight was to combine diffusion and reaction to explain patterns. Diffusion alone doesn’t create patterns—think of ink in water. Simple reactions don’t create patterns either; reactants become products, and that’s it.

Back then, everyone thought that introducing diffusion into systems would stabilize them, making them less interesting. However, Turing showed that introducing diffusion into reacting chemical systems could destabilize them and create amazing patterns.

A “reaction-diffusion system” may sound intimidating, but it’s quite simple: there are two chemicals—an activator and an inhibitor. The activator promotes its own production and creates the inhibitor, while the inhibitor suppresses the activator.

How can this be related to actual biological patterns? Imagine a cheetah with no spots. We can think of its fur as a dry forest where little fires break out. Firefighters are stationed throughout the forest and can travel faster than the fire. The fires can’t be extinguished from the middle, so firefighters outrun the fire and spray it back from the edges. This results in blackened spots surrounded by unburned trees in our cheetah forest.

Fire represents the activator chemical, which makes more of itself, while the firefighters act as the inhibitor chemical, reacting to the fire and extinguishing it. Both fire and firefighters spread throughout the forest. The key to achieving spots (and not an all-black cheetah) is that the firefighters spread faster than the fire.

By adjusting a few simple variables, Turing’s mathematical rules can create a staggering variety of patterns. The equations that produce spotted patterns like those of cheetahs can also produce stripy patterns or even a combination of the two, depending on different values within the equations. For instance, there’s a number that describes how quickly the fire chemical produces itself, as well as numbers that describe how fast each chemical diffuses. Altering these values slightly can change a spotted pattern into a stripy one.

Another factor that affects the pattern is the shape on which the pattern is created. A circle or square is one thing, but animal skins aren’t simple geometric shapes. When Turing’s mathematical rules are applied to irregular surfaces, different patterns can form in different areas. Often, when we observe nature, this predicted mix of patterns is what we see. Stripes and spots may seem like very different shapes, but they could be two versions of the same thing, with identical rules playing out on different surfaces.

Turing’s 1952 article was largely overlooked at the time, possibly overshadowed by other groundbreaking discoveries in biology, such as Watson and Crick’s 1953 paper describing the double helix structure of DNA. Or perhaps the world simply wasn’t ready to accept a mathematician’s ideas in biology. However, after the 1970s, when scientists Alfred Gierer and Hans Meinhardt rediscovered Turing patterns, biologists began to take notice. They started to wonder how these biological patterns, predicted by mathematics, are actually created in nature.

This question has proven to be surprisingly complex. Turing’s mathematics elegantly models reality, but to truly validate Turing’s theories, biologists needed to find actual morphogens—chemicals or proteins inside cells that behave as Turing’s model predicts. Recently, after decades of searching, biologists have begun to identify molecules that fit the mathematical predictions. The ridges on the roof of a mouse’s mouth, the spacing of bird feathers, the hair on your arms, and even the tooth-like denticle scales of sharks—all of these patterns are sculpted in developing organisms by the diffusion and reaction of molecular morphogens, just as Turing’s math predicted.

However, some living systems are more complex. For example, in the developing limbs of mammals, three different activator/inhibitor signals interact in intricate ways to create the pattern of fingers, resembling a binary pattern of digits.

Sadly, Alan Turing never lived to see his genius recognized. The same year he published his paper on biological patterns, he admitted to being in a homosexual relationship, which was a criminal offense in the United Kingdom at that time. Rather than face imprisonment, he underwent chemical castration treatment. Two years later, in June of 1954, at the age of 41, he was found dead from cyanide poisoning, likely a suicide. In 2013, Turing was finally pardoned by Queen Elizabeth, nearly 60 years after his tragic death.

While I don’t want to portray scientists as mythical heroes, even the greatest discoveries result from numerous failures and are built on the work of many others. They are never simply plucked from thin air. That said, Alan Turing’s work on zebra stripes and leopard spots undeniably showcases his extraordinary intellect.

The equations that produce these patterns are not easily solved with pen and paper; in most cases, we need computers to assist us. What’s remarkable is that when Alan Turing was developing these theories and studying these equations, he didn’t have access to the computers we have today.

Here are some of Alan Turing’s notes found in his home after his death. If you look closely, you’ll notice they aren’t just numbers; they resemble a secret code!

Yes, it’s like a secret code! It’s his unique code. It’s in binary, but instead of writing it out directly, he used another code that represented the binary. Alan Turing could describe the equations in a way that required millions of calculations by a computer, but he didn’t have a fast computer to do it, so it would have taken him a long time.

What has the world missed out on due to the loss of Alan Turing? It’s extremely difficult to quantify what the world has missed with his passing. Often, he struggled to communicate his thoughts to others because they were so advanced and complex, seemingly coming out of nowhere.

When you read accounts from people who knew him, they express the same sentiment: they don’t know where his ideas originated. It’s impossible to say what he could have achieved, but it would have been remarkable.

One war historian estimated that Turing and his fellow codebreakers shortened World War II in Europe by more than two years, potentially saving 14 million lives. After the war, Turing played a crucial role in developing the core logical programming that underpins every computer on Earth today, including the one you’re using to watch this video.

Decades later, his lifelong fascination with the mathematics underlying nature’s beauty has inspired entirely new questions in biology. Achieving any one of these accomplishments would be worth celebrating, but accomplishing all of them marks the work of a rare and exceptional mind—one that recognizes that the true beauty of mathematics lies not only in its ability to describe reality but also in deepening our understanding of it.

Stay curious!