How Liverpool's title win has completed a mysterious Fibonacci sequence

How Liverpool's title win has completed a mysterious Fibonacci sequence

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Liverpool FC's victory at the weekend has clinched them their second Premier League title but it also resulted in something curious – producing a strange series of numbers in the leagues' record books.

Something remarkable has just happened in English football. Liverpool FC have been crowned Premier League champions for a second time. When added to their 18 pre-Premier League titles, it means they now equal Manchester United's record of being English champions 20 times. But while fans of the club will no doubt be celebrating this moment of triumph, another astounding facet of their achievement has caught the attention of mathematicians.

Liverpool's title win has completed the opening of an exceptional set of numbers that has been 33 years in the making. The sequence emerges when we rank Liverpool alongside the other clubs that have won the Premier League since it was first formed in 1992, listing them by the number of titles won, starting with the lowest. As you can see in the table below, the number of Premier League titles goes as follows: 1, 1, 2, 3, 5, 8, 13.

Clubs who've won the Premier League, ranked by how many Premier League titles they've won, from least to most (Credit: BBC)

To the untrained eye, this sequence might not seem significant. But it will be enough to get many maths aficionados excited. They will recognise this as the Fibonacci sequence, in which each number (after the first two) is the sum of the previous two in the sequence.

The sequence can be found in an astonishing array of places – from the spirals of seeds on sunflower heads and the bracts of pinecones to family tree patterns in some species of animals.

Fibonacci sequences (sequences in the plural because starting with a different pair of initial numbers and following the rule of adding consecutive numbers to generate the next gives you a different, but related sequence) were first introduced to European science in 1202 by Leonardo of Pisa, also known by his nickname Fibonacci (meaning son of Bonaccio).

Long before Fibonacci popularised the sequences in his book Liber Abaci, however, the sequences had been known to Indian Mathematicians. They had drawn upon the sequences to help them enumerate the number of possible poems of a given length, using short syllables of one-unit duration and long syllables of two-unit duration. The Indian poet/mathematicians knew that you could make a poem of length n by taking a poem of length n-1 and adding a short syllable or a poem of length n-2 and adding a long syllable. Consequently, they figured out that to work out the number of poems of a given length you just had to add the number of poems that were one syllable shorter to the number that were two syllables shorter – the exact rule we use today to define a Fibonacci sequence.

Fibonacci sequences are often held up by mathematicians as exemplars of the beauty of mathematics. They can provide vivid visual examples of maths written into the patterns of the real world, without which many non-mathematicians can struggle to understand the elegance we see in our subject. In our over-enthusiasm to proselytise, however, there is a temptation to cast Fibonacci sequences or the golden ratio as some sort of all-encompassing natural law governing phenomena across orders of magnitude, from the spiral shapes of nautilus shells to vortices of hurricanes to the curved arms of galaxies.

In reality, although these natural features are aesthetically pleasing, very few of them conform to the rules of the Fibonacci sequence or exhibit the golden ratio. We must be careful that we don't try to shoehorn every beautiful pattern into the delicate Fibonacci glass slipper – to suggest causation and impose meaning where there is none.

Getty Images Fibonacci sequences can be found in a wide range of natural phenomena, including the heads of sunflowers (Credit: Getty Images)

Coincidence?

It's extraordinary, then, to find the Fibonacci sequence cropping up in a place as unexpected as the Premier League. When, as scientists, we spot a well-known sequence like this appearing seemingly out of the blue, we should start to ask ourselves whether it tells us anything important about the process that generated the sequence. Is there some surprising unseen process underlying Premier League title battles or is it nothing more than a cute coincidence? Just because we can see a Fibonacci sequence in something doesn't mean it is there for a reason.

Nonetheless, spotting these sorts of seeming coincidences can be extremely useful for the process of scientific discovery. In 1912, for example, Alfred Wegener noticed the apparently strange coincidence that the coastline of West Africa and the eastern coastline of South America seemed to fit together like the pieces in a jigsaw puzzle. Despite the prevailing opinion at the time, that the enormous land masses of the continents were just too big to move, Wegener proposed the only theory that reconciled his observations. Continental drift suggested that the land masses weren't rooted in place but could, ever-so-slowly, change their relative positions on the surface of the Earth.

When he published his theory in 1915, he became a laughing stock. Geologists rejected his outlandish idea, citing a lack of mechanism for moving such enormous chunks of the Earth's surface, dismissing the seemingly snug tessellation of the continents as pure coincidence. By the 1960s, however, the theory of plate tectonics – the movement of the solid mantle and crust over the Earth's surface – gave credence to Wegener's now widely accepted theories.

Alamy Leonardo of Pisa – better known as Fibonacci – drew on the work of Indian and Islamic mathematicians before him (Credit: Alamy)

The evolution of a mistake

Although coincidences can point the way to new scientific discoveries, they can also prove an obstacle to scientific progress when they appear to confirm an incorrect theory. In the early 1800s German anatomist Johann Friedrich Meckel made just such a blunder. He was a believer in scala naturae (the ladder of nature) in which humans sit above all other animals in an ordered but static hierarchy. The simplest, most primitive life forms were supposed to sit on the lowest rungs of the ladder, while the most complex and advanced beings resided on the highest. His views were hardly surprising given that this "great chain of being" was the predominant theory of the day. The now generally accepted theory of "common descent" – that multiple species descend from a single ancestral population – was only in its infancy as an idea at the time.

Meckel employed scala naturae to come up with a conjecture about his area of speciality – embryonic development. Known as recapitulation theory, he posited that, as they developed, the embryos of higher order animals (like mammals) progressed successively through forms which strongly resembled the "less perfect" animals, like fish, amphibians and reptiles, on lower rungs of the ladder. One startling, but seemingly unlikely prediction of this theory was that, as humans progressed through the "fish stage", their embryos would have gill slits.

As it happens, it was discovered in 1827 that human embryos really do have slits that resemble gills at an early stage of development. This extraordinary finding seemed to bear out Meckel's prediction and corroborate his recapitulation theory. So strong was the perceived evidence that the theory became widely accepted, and it wasn't until almost 50 years later in the 1870s that the recapitulation theory of development was finally put to bed for good as the idea of common descent started to take hold. Common descent underpins what we now know as modern evolutionary theory. It made clear that, far from undergoing a "fish stage" in the womb, the gill slits were a consequence of the fact that, sharing a common ancestor with fish, we also share much of their DNA and early developmental processes.

Sometimes coincidences can lead scientists astray, seeming to point to one conclusion when, in fact, there is an alternative explanation for the observations that is better supported by the facts.

So what does the fact that the beautiful, almost mystical Fibonacci sequence has cropped up in the data on the number of Premier League title wins mean for the beautiful game? With no plausible mechanism which could have given rise to the sequence, the answer is almost certainly nothing.

It's wonderful to have spotted this mathematical sequence in such an unlikely place, giving us a chance to reflect upon the importance of Fibonacci numbers. But a pattern doesn't always mean causality – a coincidence is sometimes just a coincidence.

And, just like Meckel's gill slits, its appearance in the Premier League records is just that – nothing more than a spectacular but ultimately misleading coincidence.

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