Bridging the Divide

Unity and Diversity: What Zebras and Sand Dunes Can Teach Us About Our Humanity
by Barb Cozzens -- December 10th, 2015

In the classic children’s book “Just So Stories”, Rudyard Kipling offered a fantastical explanation of How the Leopard Got Its Spots. (Spoiler alert) As his story goes, the leopard’s hunting partner, an Ethiopian, dabbed black marks on the exclusively “‘sandiest-yellowish-brownest” animal to help him find camouflage on rocks and leafy branches. That’s an appealing explanation for children, but a better answer may be found in a simple mathematical model, first described by mathematician Alan Turing more than six decades ago.

The Mathematics of Planet Earth

A few weeks ago, I attended the Mathematics of Planet Earth (MPE) 2013+ Workshop at the National Institute for Mathematical and Biological Synthesis in Knoxville. I know what you’re thinking and you’re not at all wrong; but as my Facebook status proudly proclaimed, I was there to get my nerd on and there’s no shame in my game.

Here’s some background: In 2013, Canadian mathematician Dr. Christiane Rousseau launched MPE to enlist mathematicians and computational scientists Untitledto help confront the issues affecting our planet and its future; topics run the gamut from climate to Ebola. It’s since evolved into a worldwide initiative, with partners from all disciplines on every continent, and the endorsement of the United Nations and many others.

The workshop opened with Dr. Rousseau sharing some examples that illustrate the power of mathematics to understand the world around us. My favorite by far: Her explanation of a single mathematical model that may account for the beautiful palette of patterns we see on planet Earth, from the spots on a leopard to vegetation on a hillside.

Turing’s Other Masterpiece

The model Dr. Rousseau spoke of was developed by British mathematician Alan Turing in 1952. If you caught the Oscar-nominated film The Imitation Game, you’re aware of Turing’s work cracking Germany’s World War II Enigma codes, potentially saving 14 to 21 million lives.

Despite these heroic efforts, Turing was later prosecuted and chemically castrated for being gay. Back in 1885, Victorian purists – a group that would put Kim Davis to shame – considered homosexuality ‘icky’ and sought to rid themselves of its scourge. They punished offenders, Oscar Wilde among them, with prison time or castration, an improvement on the pre-1891 sentence of death. Thankfully, a mere 82 years later, Britain realized the error of its ways and partially repealed the Criminal Law Amendment Act. Only a decade too late for Turing, who at the age of 42, died from self-inflicted cyanide poisoning, precipitated by the shame and physical effects of castration.

[Author’s notes: a) Not what I’d call a hero’s welcome and b) check out Ep. 3/Season 4 of PBS’ Call the Midwife for a beautiful episode on this topic (shout out to Kleenex™ for helping me through it)]

Two years before his death Turing wrote “The Chemical Basis of Morphogenesis,” his first and only paper on biology. In this paper, Turing theorized that any repeating natural pattern could be created by a reaction between two substances. The substances, then-unknown elements Turing referred to as morphogens (“shape formers”), interact and spread through space, spontaneously self-organizing into patterns. He called this mathematically-simple, yet elegant principle a reaction-diffusion (RD) model.

Simple enough, right?

Renowned Scottish mathematical biologist James D. Murray offered this ‘intuitive’ analogy: Imagine a field of dry grass, populated by grasshoppers. Fires start at random intervals, prompting the grasshoppers to sweat profusely. Okay, let me stop right there, because absolutely no part of that is intuitive. Apologies to Dr. Murray.

Turing Patterns of Nature

Ripples in Sand Dunes and Zebra Stripes are the result of Turing mechanisms

Sand dunes ripples and zebra stripes can be replicated using Turing’s RD model.

The essence of Turing’s model is that patterns form spontaneously from the interaction of two elements, both of which can spread through space. But rather than spreading evenly, they do so at different speeds. One, an “activator”, starts by making more of itself. At some point, it switches on the second element, an “inhibitor”, which slows down production of the activator. The inhibitor always moves at a faster pace, so it eventually catches up with and switches off the activator. Each iteration of this process would make, for example, one stripe.

In Murray’s example, fire serves as the activator and the grasshoppers – specifically their sweat – acts as the inhibitor. Spots form where moisture from the grasshopper’s sweat prevents the grass from burning. Varying parameters such as the rates of spread and decay, or the speed by which the inhibitor shuts down the activator, changes the pattern to create spots, stripes or swirls.

The intricate patterns on seashells, stripes on angelfish, waves in sand dunes and all manner of animal coat patterns, from zebra stripes to the jaguar’s broken-ring rosettes, can be replicated using Turing’s RD model. And, more recently, scientists have linked Turing-type mechanisms to the development of biological structures, including the spacing between hair follicles in mice and feather buds in chicks, the formation of palatal ridges in a mouse’s mouth, and the embryonic development of branched lungs.

At the ecological level, Turing patterns have been identified in predator-dependent models, with prey functioning as activators and predators as inhibitors. The result: patches – or spots – of high prey density surrounded by areas of low prey density.

Vegetative patterns in the semi-arid W National Park in Niger

Vegetative patterns in the semi-arid W National Park in Niger

In semi-arid ecosystems, which support over a third of the world’s population, Turing patterns arise from the interaction of water, the inhibitor, and vegetation, the activator. Modeling and understanding these vegetation patterns may help predict when a semi-arid ecosystem is approaching collapse, and thus help shape responses to climate change.

Turing’s Legacy: Unity in Diversity

Turing’s fascination with the underlying logic of the living world led him to believe that nature’s complex and richly different patterns could be explained by a single mathematical equation. Tragically, he died before seeing his vision validated. Now, more than a half century later, it seems Turing cracked another, even more enigmatic code.

The beauty of blogging is I get to share my personal opinion (FYI: get used to it). So here it is: Turing was gifted with an ability see beyond what is different – patterns and shapes – and find that exquisitely simple, unifying truth. And yet, in a Shakespearean-worthy twist of irony, he was persecuted for being ‘different’.

A mathematical ecologist friend shared her thoughts on Turing:

I have always loved the parts of math biology that can explain the seemingly chaotic patterns around us with simple mathematical models. We fear diversity, we fear what seems to be unorganized, we fear different. And the man who embodied so many of these fears was the one to first see that a single, give-and-take, reaction-diffusion model can explain the entire outcome.

With that, the next time you see a leopard, or even your tabby cat, I hope you think of Alan Turing. In the not-so distant past, we let fear and intolerance rob humanity of one of its best and brightest minds – one of the very minds that might one day have contributed to its salvation.

History will no doubt place Alan Turing among science’s elite. But it’s in the words of Mahatma Gandhi that we may find Turing’s true legacy:

“Our ability to reach unity in diversity will be the beauty and the test of our civilization.”

More information

For a more in-depth look at Turing’s work, Philip Ball wrote this commentary to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.



  1. Dave Grace
    Dec 10, 2015

    Bravo & Encore

  2. Lynn Keller
    Jan 1, 2016

    Awesome and eloquent, Barb. Keep ’em comin’.

  3. Dan Taylor
    Jan 5, 2016

    A fascinating and enlightening piece. What a wonderful job of weaving together the life and work of a visionary mathematician and a poignant message on humanism and social activism. Well done.

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