Plus Magazine Monthly Column – June 2017 – The ocellated lizard and other stories

By on 10/06/2017
Some visualisations of mathematical objects, like this one of the exceptional Lie group E8, are undoubtedly beautiful. But is maths by itself beautiful? (Image Jgmoxness, CC BY-SA 3.0.)

Plus http://plus.maths.org is a free online magazine about mathematics aimed at a general audience. It is part of the Millennium Mathematics Project, based at the University of Cambridge and our aim is to open a door onto the world of maths for everyone. We run articles, videos and podcasts on all aspect of mathematics, from pure maths and theoretical physics to mathematical aspects of art, medicine, cosmology, sport and more. Plus has a news section, covering news from the world of maths as well as the maths behind the mainstream news, reviews of books, plays and films, as well as puzzles for you to sharpen your wits.

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Beauty in the brain of the beholder

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty.” Most mathematicians would agree with this statement by Bertrand Russell. There’s something undeniably aesthetic about a succinct equation or elegant logical argument.

But how does mathematical beauty compare to the beauty we ascribe to, say, a work of art? In a recent Plus article  Josefina Alvarez explores a study conducted by two neurobiologists and a mathematician that was designed to answer this question. The team used functional MRI to look at the brains of mathematicians while they were looking at mathematical formulas. It turns out that beautiful equations do indeed activate the same areas in the brain that have been associated with the appreciation of other kinds of visual, audio or moral beauty.

Some visualisations of mathematical objects, like this one of the exceptional Lie group E8, are undoubtedly beautiful. But is maths by itself beautiful? (Image Jgmoxness, CC BY-SA 3.0.)

Some visualisations of mathematical objects, like this one of the exceptional Lie group E8, are undoubtedly beautiful. But is maths by itself beautiful? (Image Jgmoxness, CC BY-SA 3.0.)

But is all mathematics equally beautiful? Obviously not! As part of the study participants were presented with sixty mathematical formulas and asked to rank them according to three categories; ugly, neutral and beautiful. The expression that was most consistently ranked as beautiful was Euler’s equation,

$$\exp{i\pi} + 1= 0.$$

And the one most consistently ranked as ugly by the mathematicians was the following, extremely convoluted, formula representing the reciprocal of $\pi$ as an infinite sum:

$$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!   (1103 + 26390k)}{(k)!^4 396^{4k}}.$$

It appears there is a connection between the simplicity of a mathematical expression and its perceived beauty.

Spotting lizards

Another thing that often amazes people is the “unreasonable effectiveness” of maths: the fact that it appears to describe nature better than any other language or formalism we have developed. This brings us swiftly to the subject of lizards.

The ocellated lizard is a beautiful creature. When it’s young it’s brown with white dots, but as it gets older its scales become black and green. Individual scales continue to switch colour, until the simple polka dot pattern turns into an intricate black-and-green labyrinth. It’s an amazing transition — but how does the lizard do it?

Lagarto ocelado 1

The answer seems to come from some relatively straightforward maths. The lizard’s scales are roughly hexagonal, so imagine a honeycomb pattern in which each individual hexagon is either black or green. Now imagine that periodically each hexagon looks at its six neighbours and changes its colour depending on what the colours of these neighbours are. For example, in the pictures below each hexagon changes its colour if four or more of its neighbours are not of its own colour. As time goes by, scales keep switching colours and new patterns evolve. In our example, which shows three time steps, small islands of one colour surrounded by the other colour gradually disappear, until only bands of colours remain.

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This kind of set-up — a pattern in which cells update their colour at each time step depending on the colour of their neighbours at a previous time step — is called a cellular automaton. Because cellular automata are good at creating patterns, they’ve been used to simulate processes that happen in nature (see, for example, this article), but so far nobody has found a biological system that actually is a cellular automaton.

It seems like the ocellated lizard might provide the first example. In a recent study a team of mathematicians and geneticists (including the Fields medallist Stanislav Smirnov) watched three male lizards as they grew from little hatchlings into three to four-year-old adults. Carefully counting lizard spots, the team tried to find out whether a scale flipping colour at a particular time depends on the colours of its direct neighbours at a previous time step. Their work strongly suggests that it does, which means that the lizard’s scales do indeed behave like a cellular automaton, albeit with a slight difference to our example above.  Scales aren’t sure to change colour as soon as their neighbours display a certain colour configuration. Instead, they change colour with a certain probability, which depends on the neighbouring colours: the more neighbours of a scale have the same colour as the original scale, the more likely the original scale is to change colour. The lizard’s scales seem to form what’s called a probabilistic cellular automaton.

But how do the scales do that? After all, individual skin cells don’t know what scale they’re part of, they don’t have binoculars to see what colour a neighbouring scale is, or a mind to decide whether to switch colour or not. The authors of the study suggest that the lizard’s cellular automaton arises from a well-known mechanism for creating skin patterns that was first suggested by the famous mathematician and code breaker Alan Turing in the 1950s. Although Turing’s model has nothing to do with cellular automata, a cellular automaton can arise from it if the thickness of the lizard’s scales varies. Find out more in this article.

Making history and mathematical magic!

The ocellated lizard might have made history with its cellular automoton decoration, but you too can make history without changing colour at all.  Just shuffle a deck of cards – chances are you’ll produce an order of cards that has never ever occurred before in the whole, long history of the Universe.

Although they can’t guarantee that your shuffled pack is a universal first – Mathematician Ricardo Teixeira, and his daughter Gisele, use probabilities to explain why it is almost a certainly.

For a standard 52 pack of cards, there are 52 possibilities for the first card, 51 for the second, and so on. Hence the total number is:

$$52 \times 51 \times 50 \times 49 \dots \times 2 \times 1 = 8.07 \times 10^{67}.$$

Therefore, the probability of producing the order of your shuffled pack of cards is

$$\frac{1}{(8.07 \times 10^{67})}$$

To show just how incredibly small this probability is, imagine 10 billion people each with a pack of cards, each shuffling their pack once per second.  Then after the first second they would have produced at most $$10^{10}$$ (10 billion) different combinations of cards (assuming each of the 10 billion people produced a different order of cards from their first shuffle). After the first minute, there would be 10 billion times 60 new combinations, that’s $$10^{10}\times 60 = 6 \times 10^{11}$$. The pattern continues like this:

gatti2

But this number is less than a billionth of a billionth of a billionth of a billionth of 1 percent of the number of possible card sequences,$$52 \times 51 \times 50 \times 49 \dots \times 2 \times 1$$. Thus even after 14 billion years of 10 billion people shuffling a uniquely ordered pack every second, still only a tiny fraction of all the possible combinations of cards would have been produced!

This is why you can be pretty sure that any magician isn’t using just dumb luck to guess your card in a magic trick. They probably aren’t using magic either.   To see more, including how to impress with the magic trick in the video below, read the full article.

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About Marianne Freiberger

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