Turing Patterns

What is a Turing Pattern?

In 1952 famous mathematician Alan Turing wrote down an idea about how humans and animals grow. Alan imagined that there were chemicals inside our bodies that tell us how to grow. He imagined that these chemicals create patterns on the skin of animals - like cheetahs and zebras. Alan wrote his ideas into his favourite language, the language of maths! Maths isn't just about numbers, we can also use it to create patterns, like art.

Did you know that animals and humans grow from a tiny ball of cells and become different shapes? Have you ever wondered why the hairs on our arms grow in a spotted pattern or why our fingerprints have that striped pattern? Alan Turing’s mathematics tells us how these patterns can appear. Lots of scientists are using Turing Patterns to help explain things in nature.

Read the rest of this page to find out more!

Turing Patterns that look like parts of animals and humans

James D Murray used Turing’s mathematics to recreate animal coat patterns. Murray showed that Turing patterns explain why we see animals with spotted bodies and stripy legs and tails!
Murray showed that as tails and legs get thinner, spots turn to stripes.
James Murray as a Turing Pattern (Woolley, Krause & Gaffney 2021)
Even though the patterns created by the maths look like patterns in real life, it is hard to know whether or not Alan really was correct about big animals like cheetahs and zebras. That's because scientists can’t watch the chemicals inside baby animals as they grow inside their mothers.
A Turing Pattern at the top and a real jaguar at the bottom. (Painter 2000)
The pattern around this fish's eye (left) can be made using Turing's maths (right). (Kondo 2010)
We can make a pattern like the human brain (left) using Turing's maths (right). (Cartwright 2002)

Turing Patterns where scientists have found the chemicals!

Scientists can’t search for the chemicals inside big animals, but they can study smaller animals much easier. Here’s a few places where scientists have managed to find the chemicals that Turing imagined inside some real animals. First, the zebra fish:
A zebra fish. Scientists can remove the stripes to see what happens when they grow back! (Watanabe 2014)
At the top are photographs that show the different patterns that appear on the fish with different amounts of chemicals. Below shows the Turing Pattern created from the maths. Its amazing that they are almost the same!

Scientists have also found that the chemicals described by Turing make hair and bird feathers grow in a spotted pattern. Have a look at these photographs and mathematical patterns:

A photograph of the hairs on a mouse and the Turing Pattern from the maths below. (Sick 2006)
The pattern of feathers on a quail and pheasant. (Bailleul 2016)
The pattern of how whiskers grow is a Turing Pattern! (Krause 2018)

Turing Patterns in other parts of nature

The maths that Alan Turing used to describe the patterns on animals can be seen in lots of other parts of nature.
Complicated patterns on shells (left) can be created from Turing’s maths too (right). (Meinhardt 1995)
When ants go to battle against each other, they create graveyards. Scientists think that the pattern of these graveyards is a Turing Pattern too! (Theraulaz 2002)

Turing Patterns at the University of Sheffield

At the University of Sheffield, we use Alan Turing’s maths to figure out even more amazing facts about nature. Did you know that sharks have tiny teeth all over their skin called denticles? We can study some sharks as they grow in a transparent egg case! Scientists at the University use maths to show that the way these denticles grow is a spotted pattern and that chemicals inside the shark control this pattern, just as Alan Turing imagined.
Scientists at the University also use Turing's maths to look at the patterns of the scales on the feet of chickens and reptiles.

This photograph is the foot of a chameleon!
We also use maths to create more complicated patterns. Like the skin of this angelfish.
Have you ever seen inside a cat's mouth? They have a striped pattern on the roof of their mouth that's much more detailed than ours. Scientists at the University of Sheffield think this is a Turing Pattern too! Alan Turing’s maths doesn’t just describe parts of animals but also patterns on the earth. Imagine a drone flying high in the sky and filming forests and mapping out where animals might be. Some animals live in small territories and we can using Turing's ideas to find the patterns created by the animals. Instead of chemicals, the maths describes animal behaviours, like searching for food!

Scientific References

  • Cartwright, J.. “Labyrinthine Turing pattern formation in the cerebral cortex.” Journal of theoretical biology 217 1 (2002): 97-103 .
  • Cooper, R.L., Lloyd, V.J., Di-Poï, N. et al. Conserved gene signalling and a derived patterning mechanism underlie the development of avian footpad scales. EvoDevo 10, 19 (2019). https://doi.org/10.1186/s13227-019-0130-9
  • R.L. Cooper, A.P. Thiery, A.G. Fletcher, D.J. Delbarre, L.J. Rasch, G.J. Fraser. An ancient Turing-like patterning mechanism regulates skin denticle development in sharks. Sci. Adv., 4 (11)
  • Economou. A. D, Monk. N. A. M, Green. B. A. J. (2020) Perturbation analysis of a multi-morphogen Turing reaction-diffusion stripe patterning system reveals key regulatory interactions. Development 2020 147: dev190553 doi: 10.1242/dev.190553
  • Ellison, N, Hatchwell, BJ, Biddiscombe, SJ, Napper, CJ, Potts, JR. Mechanistic home range analysis reveals drivers of space use patterns for a non-territorial passerine. J Anim Ecol. 2020; 00: 1– 14. https://doi.org/10.1111/1365-2656.13292
  • Krause. A. L, Klika. V, Woolley. T. E, and Gaffney. E.A. (2018) Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems. Phys. Rev. E 97, 052206
  • Kondo, S. and Miura, T. (2010) Biological pattern formation reaction-diffusion model as a framework for understanding biological pattern formation. Science, 329, 1616-1620.
  • Meinhardt, H. (1995) The Algorithmic Beauty of Sea Shells. Springer.
  • Painter. K.J, Maini. P.K. Othmer. H.G. (2000) Chemotactic response to multiple signalling cues. J. Math. Biol., 41, 285-314 120 https://link.springer.com/ article/10.1007/ s002850000035
  • Sick, S., Reinker, S., Timmer, J. and Schlake, T. (2006). WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism. Science 314, 1447-1450. doi:10.1126/science.1130088
  • Theraulaz G, et al. (2002) Spatial patterns in ant colonies. Proc. Natl Acad. Sci. USA. 99, 9645–9649.doi:10.1073/pnas.152302199
  • Watanabe. M, Kondo. S. (2014) Is pigment patterning in fish skin determined by the Turing mechanism? https://doi.org/10.1016/j.tig.2014.11.005