- The living world is a universe of shapes and patterns, beautiful, complex, and sometimes strange, and beneath all of them is 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 in between.

It's a code written, not in the language of DNA, but in math.

Can simple equations really explain something as messy and unpredictable as the living world?

How accurately can mathematics truly predict reality?

Could there really be one universal code that explains all of this?

(compelling music) Hey, smart people, Joe here.

What color is a zebra?

Black with white stripes, or white with black stripes?

This is not a trick question.

The answer is black with white stripes, and we know that, because some zebras are born without their stripes.

It might make you wonder, why do zebras have stripes to begin with?

A biologist might answer that question like this, the stripes aid in camouflage from predators, and that would be wrong.

The stripes' actual purpose is most likely to confuse bloodthirsty, biting flies.

(flies buzzing) Yup, but that answer, it really just tells us what the stripes do, not where the stripes come from, or why patterns like this are even possible.

Our best answer to those questions?

Well, it 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 that we see in nature, ranging from stripes, to spots, to labyrinth-like waves, and even geometric mosaics, all now known as Turing patterns.

Most people know Alan Turing as a famous wartime codebreaker, and the father of modern computing.

You might not know that many of the problems that most fascinated him throughout his life were, well, about life, about biology, but why would a mathematician be interested in biology in the first place?

- That's a really good question.

I'm Dr. Natasha Ellison, and I'm from the University of Sheffield, which is in the UK.

I think so many mathematicians are interested in biology, because it's so complicated, and there's so much we don't know about it.

If you think about a living system, like a human being, there's just so many different things going on, and really, we don't know everything.

- The movements of animals, population trends, evolutionary relationships, interactions between genes, or how diseases spread, all of these are biological problems, where mathematical models can help describe and predict what we see in real life.

But mathematical biology is also useful for describing things that we can't see.

What do you say when people ask, "Why should we even care about math in biology?"

- Why should we care about mathematics in describing biology?

And the reason is because there's things about biology that we can't observe.

- We can't follow every animal all the time in the wild, or observe their every moment.

It's impossible to measure every gene and chemical in a living thing at every instant.

Mathematical models can help make sense of these unobservable things, and one of the most difficult things to observe in biology is the delicate process of how living things grow and get their shape.

Alan Turing called this morphogenesis, the generation of form.

In 1952, Turing published a paper called The Chemical Basis of Morphogenesis.

In it was a series of equations describing how complex shapes like these can arise spontaneously from simple initial conditions.

According to Turing's model, all it takes to form these patterns is two chemicals spreading out the same way atoms of a gas will fill a box, and reacting with one another.

Turing called these chemicals morphogens, but there was one crucial difference.

Instead of spreading out evenly, these chemicals spread out at different rates.

- So the way that we create a Turing pattern is with some equations called reaction-diffusion equations, and usually, they describe how two or possibly more chemicals are moving around, and reacting with each other.

So diffusion is the process of sort of spreading out.

So if you can imagine, I don't know, if you had a dish with two chemicals in it, they're both spreading out across the dish, they're both reacting with each other, this is what we reaction-diffusion equations are describing.

- This was Turing's first bit of genius, to combine these two ideas, diffusion and reaction, to explain patterns, because diffusion, on its own, doesn't create patterns.

Just think of ink in water.

Simple reactions don't create patterns, either.

Reactants become products, and that's that.

- So everybody thought back then that if you introduce diffusion into systems, it would stabilize it, and that would basically make it boring.

What I mean by that is you wouldn't see a lovely pattern.

You'd have an animal just one color.

But actually, Turing showed that when you introduce diffusion into these reacting chemical systems, it can destabilize, and form these amazing patterns.

- A reaction-diffusion system may sound intimidating, but it's actually pretty simple.

There are two chemicals, an activator, and an inhibitor.

The activator makes more of itself, and makes inhibitor, while the inhibitor turns off the activator.

How can this be translated to actual biological patterns?

Imagine a cheetah with no spots.

We can think of it's fur as a dry forest.

In this really dry forest, little fires break out, but firefighters are also stationed throughout our forest, and they can travel faster than the fire.

The fires can't be put out from the middle, so they outrun the fire, and spray it back from the edges.

We're left with blackened spots, surrounded by unburned trees in our cheetah forest.

Fire is like the activator chemical.

It makes more of itself.

The firefighters are the inhibitor chemical, reacting to the fire, and extinguishing it.

Fire and firefighters both spread, or diffuse throughout the forest.

The key to getting spots, and not an all black cheetah is that the firefighters spread faster than the fire.

And by adjusting a few simple variables like that, Turing's simple set of mathematical rules can create a staggering variety of patterns.

- These equations that produce spotted patterns like cheetahs, the exact same equations can also produce stripy patterns, or even a combination of the two, and that depends on different numbers inside the equations.

For example, there's a number that describes how fast the fire chemical will produce itself.

There's a number that describes how fast the fire chemical would diffuse, and how fast the water chemical would diffuse as well, and all of these different numbers inside the equations can be altered very slightly, and then we'd see instead of a spotted pattern, a striped pattern.

- And one other thing that affects the pattern is the shape you're creating the pattern on.

A circle or square is one thing, but animals skins aren't simple, geometric shapes.

When Turing's mathematical rules play out on irregular surfaces, different patterns can form on different parts, and often when we look at nature, this predicted mix of patterns is what we see.

We think of stripes and spots as very different shapes, but they might be two versions of the same thing, identical rules playing out on different surfaces.

Turing's 1952 article was largely ignored at the time, perhaps because it was overshadowed by other groundbreaking discoveries in biology, like Watson and Crick's 1953 paper describing the double helix structure of DNA, or perhaps, because the world simply wasn't ready to hear the ideas of a mathematician when it came to biology.

But after the 1970s, when scientists Alfred Gierer and Hans Meinhardt rediscovered Turing patterns in a paper of their own, biologists began to take notice, and they started to wonder.

Creating biological patterns using mathematics may work on paper, or inside of computers, but how are these patterns actually created in nature?

It's been a surprisingly sticky question to untangle.

I mean, Turing's mathematics, they simply and elegantly model reality, but to truly prove Turing right, biologists needed to find actual morphogens, chemicals or proteins inside cells that do what Turing's model predicts.

And just recently, after decades of searching, biologists have finally begun to find molecules that fit the math.

The ridges on the roof of a mouse's mouth, the spacing of bird feathers, or the hair on your arms, even the tooth-like, dentical 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.

But as simple and elegant as Turing's math is, some living systems have proven to be a bit more complex.

In the developing limbs of mammals, for example, three different activator inhibitors signals interact in elaborate ways to create the pattern of fingers, stripe-like signals, alternating on and off, like ones and zeros, 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 at the time, was a criminal offense in the United Kingdom.

Rather than go to prison, he submitted to chemical castration treatment with synthetic hormones.

Two years later in June of 1954, 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.

Now, I don't like to make scientists sound like mythical heroes.

I mean, even the greatest discoveries are the result of failure after failure, and are almost always built on the work of many others.

They're never plucked out of the ether, and just put in someone's head by an angel of genius.

But that being said, Alan Turing's work decoding zebra stripes and leopard spots leaves no doubt that he truly was a singular mind.

- The equations that produce these patterns, we can't easily solve them with pen and paper, and in most cases, we can't at all, and we need computers to help us.

So what's really amazing is that when Alan Turing was writing these theories, and studying these equations, he didn't have the computers that we have today to be able to understand when the patterns form, and to see those patterns really quickly.

So this here is some of Alan Turing's notes that were found in his house when he died.

You'll notice that they're not actually numbers.

- Wow, it's like a, it's like a secret code.

- Yeah, it's like a secret code.

It's his secret code, but it's in binary, actually, but instead of writing in binary out, because you've got the five digits, he had this sort a code that kind of coded on the binary.

So Alan Turing could describe the equations in this way that required millions of calculations by a computer, but he didn't really have, you know, well, he didn't have a fast computer to do it, so it would have taken him absolutely ages.

- What has the world lost out on by the fact that we've lost Alan Turing?

- It's extremely hard to describe what the world's missed out on with losing Alan Turing, because so often, he couldn't communicate his thoughts to other people, because they were so far ahead of other people, and they were so complicated.

They seemed to come out of nowhere sometimes.

But when you read accounts of people who knew him, they were saying the same thing.

We don't know where he got this idea from.

So what he could have achieved, I don't think anyone could possibly say.

I have no idea where we would have got to, but it would have been brilliant.

- One war historian estimated that the work of Turing, and his fellow codebreakers shortened World War II in Europe by more than two years, saving perhaps 14 million lives in the process.

And after the war, Turing was instrumental in developing the core logical programming at the heart of every computer on Earth today, including the one you're watching this video on, and decades later, his lifelong fascination with the mathematics underlying nature's beauty has inspired completely new questions in biology.

Doing any one of these things would be worth celebrating.

To do all of them is the mark of a rare and special mind, one that could see that the true beauty of mathematics is not just its ability to describe reality.

It's to deepen our understanding of it.

Stay curious.