# How The Leopard Got His Spots

###### An article on Professor Ian Stewart's Christmas 2012 mini lecture

Published December 2012

What’s the link between Christmas trees and Fibonacci numbers? Why do leopards have spotted bodies and tails, while cheetahs have spotted bodies and striped tails? In this five-minute festive lecture, Professor Ian Stewart looks at patterns in nature, focusing on Alan Turing’s work as a pioneer of mathematical biology.

“Christmas trees, pine cones… I want to talk to you about some Christmassy mathematics,” begins Professor Ian Stewart, holding up a pinecone. If you look closely, he explains, you will see eight spirals in one direction and 13 in the other. “Now those are called Fibonacci numbers, they’re very famous in mathematics, they go back to 1202 when Leonardo of Pisa – who was later given the nickname Fibonacci – wrote a textbook of arithmetic. These numbers came up in a rather silly problem about rabbits. But people quickly discovered that they seemed to occur very commonly in the plant kingdom. If you count the numbers of petals on flowers, you look at the spacing of branches on trees, you look at pineapples, and you look at pine cones, you keep seeing these Fibonacci numbers.”

Over about a period of 200 years, mathematicians and biologists between them began to put together a pretty good description of the geometry of these numbers, but they still hadn’t completely explained where they come from. Then along came Alan Turing.

##### Alan Turing and Friesian Cows

“Turing was interested in these Fibonacci numbers and patterns on plants,” says Prof Stewart, “in his unpublished papers there’s a very extensive piece of unfinished work about it, but he did publish on a related problem, which is markings on animals.” Turing had a particular interest in Friesian cows , says Prof Stewart, due to their black and white, patchy, dappled pattern. “Turing came up with a mathematical theory of these patterns based on the idea that there must be some kind of chemical, which he called a morphogen… these morphogens are chemicals that are somehow produced in an embryo animal and they react together to create other chemicals, and all of these chemicals diffuse, spread sideways across the surface.”

If you do the mathematics of this, says Prof Stewart, you find this reaction-diffusion system (also referred to as a ‘Turing system’) crops up very commonly with waves. “Think of waves coming up the beach – you have a whole series of parallel, straight waves coming up the beach, and if you took a snapshot of that in an instance of time, it looks like stripes. The top of the waves look like the dark stripes on a tiger, and the troughs of the waves could be the orange patches and white in-between. These Turing equations have these wave patterns – and, if instead of one set of waves, you have two sets of waves that cross each other, then you get spots. So the big difference between tigers and leopards is the tiger has one set of waves, the leopard has two.”

##### Professor James Murray and Big Cats

Mathematical biologist Professor James Murray made more discoveries when he applied Alan Turing’s theory to markings on big cats. “He realised that something quite interesting follows from this description of chemical waves,” explains Prof Stewart, “which is that a striped animal cannot have a spotted tail but a spotted animal can have a striped tail – so the tiger has striped body, striped tail; the leopard has a spotty body and a spotty tail and a cheetah has a spotty body and a striped tail but you don’t find a big cat – or indeed, very much any other kind of animal – where you have a striped body and a spotted tail. And the reason is, if you have room for two sets of waves to form spots on the tail, there’s even more room on the body for two sets of waves, so you will tend to get spots on the body as well.”

For the complete lecture, which includes information on sonic hedgehogs and morphogens in plants, watch the video above.

This is one of five Christmas mini-lectures, produced as part of the University of Warwick’s 2012 online Christmas greeting. You can watch all five of the videos, on subjects ranging from the Medieval Christmas to Santa’s special rocket booster, here: http://www2.warwick.ac.uk/christmas2012

Professor Ian Stewart has honorary doctorates from Westminster, Louvain, Kingston, and the Open University. He is an Emeritus Professor and Digital Media Fellow in the Mathematics Department at Warwick University, with special responsibility for public awareness of mathematics and science.

He has published more than 80 books including recent popular science books Why Beauty is Truth, Letters to a Young Mathematician, The Mayor of Uglyville’s Dilemma, Figments of Reality, The Magical Maze, Life’s Other Secret, the UK bestselling series The Science of Discworld I, II, and III (with Terry Pratchett and Jack Cohen), the US bestseller Flatterland, What Shape is a Snowflake? and the bestselling Professor Stewart's Cabinet of Mathematical Curiosities. A sequel, Professor Stewart's Hoard of Mathematical Treasures, was published in 2009.

His present field of research is the effects of symmetry on dynamics, with applications to pattern formation and chaos theory in areas including animal locomotion, fluid dynamics, mathematical biology, chemical reactions, electronic circuits, computer vision, quality control of wire, and intelligent control of spring coiling machines.