By Kit Yates
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 league's 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.
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.
Hidden in the sequences is another important and related mathematical mainstay – the golden ratio. As the terms in a Fibonacci sequence get larger, the ratio of each term to the one preceding it gets closer and closer to the golden ratio – approximated to 1.61803 by the first few places in its decimal expansion. The golden ratio is hypothesised to govern the arrangement of leaves on the stem of some plant species and supposedly leads to aesthetically pleasing results when applied in art, architecture and music.
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.
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.
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