Fibonacci Sequence: Mathematicians Spot Something Odd After Liverpool Win Premier League

That’s an unlikely sentence to read on a science website, but there’s a reason we’re talking about it. Despite being one of the top clubs in England for decades (full disclosure, I am a fan), this was only their second title victory since the Premier League was founded in 1992. 

Mathematicians noticed something unusual had been completed, after this result. With two wins to Liverpool’s name, the top seven Premier League title wins make a very familiar pattern.

Football club	Number of Premier League title wins
Blackburn Rovers	1
Leicester City	1
Liverpool	2
Arsenal	3
Chelsea	5
Manchester City	8
Manchester United	13

While football fans may notice “Manchester = unusually good at football”, any mathematician or math enjoyer will recognize that these win totals spell out the Fibonacci sequence, a sequence of numbers that seems to pop up in all sorts of surprising places.

In terms of math, the sequence isn’t too hard to get your head around. In short, each number is the sum of the two numbers that preceded it. So 0+1 = 1, 1+1 = 2, 2+1 =3, 3+5 = 8, 5+8 = 13, 8+13 = 21, and so on for infinity. But mathematicians are particularly interested in it for its habit of showing up everywhere, and in seemingly unrelated areas.

This sequence, first spotted by medieval Indian scholars attempting to investigate patterns in Sanskrit poetry, was noticed by Italian mathematician Leonardo Bonacci, known as Fibonacci, while studying the breeding patterns of rabbits.

“A common reproductive assumption was that each pair of rabbits begets another pair every month. Start with a single rabbit pair, and successive populations will then follow the sequence 1, 2, 4, 8, 16, 32, 64, 128, 256 and so on – that is, get multiplied by a monthly ‘growth ratio’ of 2,” Manil Suri, Professor of Mathematics and Statistics at the University of Maryland, Baltimore County, explains in a piece for The Conversation.

“What Fibonacci observed, though, was that rabbits spent the first cycle reaching sexual maturity and only began reproducing after that. A single pair now gives the new, slower progression 1, 1, 2, 3, 5, 8, 13, 21, 34… instead. This is the famous sequence named after Fibonacci; notice that each population turns out to be the sum of its two predecessors.”

The sequence is strongly linked to the Golden Ratio, phi.

“How does phi show up amid all these randy rabbits? Well, progressing through the sequence, you see that each number is about 1.6 times the previous one,” Suri continues. “In fact, this growth ratio keeps getting closer and closer to 1.618…. For instance, 21 equals about 1.615 times 13, and 34 equals about 1.619 times 21. This means the rabbits settle down to reproducing with a growth ratio that is no longer 2, but rather, gets closer and closer to the Golden Ratio.”

The sequence pops up in other areas of nature, for instance in the branches of trees and the spiral patterns of intolerable vegetable the cauliflower, though rarely does it conform perfectly to the rule.

In nature, the appearance of these numbers can be explained if you dig a little deeper.

“Think about the leaves on a plant. The plant’s energy comes from the Sun, so its goal as it grows is to maximize its leaves’ exposure to the sunlight. The obvious way to do that is to make sure new leaves grow a little way round the stem from the previous one – but how far round should it go?” IFLScience resident mathematician Dr Katie Spalding explains.

“Well, let’s try some examples. Halfway round won’t do; by the time you grow a third leaf, it will be directly underneath the first one, and won’t be able to see the sun. The same is true for a third, or a quarter, or a fifth of the way around – in fact, any rational fraction of the way around will eventually mean one leaf is entirely in the shade of another.

The answer, therefore, must be to go for an irrational fraction of a revolution – and the best of all must be the most irrational fraction. As we’ve seen, the best way to reach that particular value – after all, it’s physically impossible to do so exactly – is to use the Fibonacci numbers.”

So why is the Fibonacci sequence showing up in the Premier League of all places? If you wanted to get real cranky, you could try to come up with a plausible mechanism, for instance guessing that it has something to do with optimization of the Premier League, with resources (i.e. money) going to the top team, and less to the subsequent team and so on, and the win pattern emerges as a result.

But, while footballers are not immune to math, that is very cranky indeed, and almost certainly far from correct. It is much more plausible to say that these numbers merely showed up by coincidence, and are noticed by humans with our pattern-hungry eyes.

“While there are many spiral shapes that occur from purely physical, non-biological processes in nature – from whirlpools and vortices that form in bodies of water to the aerial shapes of hurricane clouds and clear lanes – none of these spirals are Fibonacci-like when it comes to the actual mathematical details of their structures on a sustained basis,” astrophysicist and science communicator Ethan Siegel points out in an article for Big Think.

“You may be able to take a ‘snapshot’ where one or more of the features exhibits ratios that are consistent with the ratios found in the Fibonacci sequence for a particular moment, but those structures don’t endure and persist.”

If any of those top teams wins the Premier League next year, for example, the Fibonacci sequence will be ruined. We would have to wait a very long time for the next step of the Fibonacci sequence to show up again, and we wouldn’t bet on it happening, especially when it’s easier to just place a large bet on one of the teams from Manchester grabbing the title again.

[H/T: BBC]