Φ

Phi is one of very few numbers to have three ‘names’ of sorts; the first is, of course, phi (from the Greek letter Φ, pronounced ‘fee’), and the second is its numerical representation, 1.618 (to four significant figures; the number itself is equal to (1+√5)/2). The third comes courtesy of Dan Brown,  semi-conspiracy theorists and lots of gullible people around the world, and generally comes in a form similar to ‘SERIOUSLY IT’S ACTUALLY A THING DON’T GO AWAY PLEASE’.

Y’see, phi is a number with a great deal of myths, or at least half-truths, surrounding it, which lead a lot of people who don’t do enough research into things to believe it all holds a vast array of semi-magical properties, ranging from molluscs to architecture. Many of this myths, some of which shall be explored later, found their way into Chapter 20 of The Da Vinci Code, which (some might say unfortunately) went on to be a bestseller. Dan Brown is an entertaining author, but a great deal of his work is based around these sort of half truths. This is hardly something that only he is guilty of as an author, but unfortunately a habit of including a section named ‘Facts’ at the start of his books and a tendency to at least start from a position of truthfulness has lead a few too many people to think that far too much of what he says is true. Hence why large portions of people get very, very angry at him, and why phi is rarely a concept taken seriously within intellectual circles.

Anyway; back to the number itself. Phi’s unique property as a number is, seemingly, innocuous enough; if you subtract 1 from it, and then divide one by that number, you return to 1.618 (or, to put it another way, 1/0.618=1.618). Go find a calculator and try it if you want; if you set it up to perform this function [1/(1-Ans)=Ans], you can start from any number above 1 and should end up at phi after a few iterations.

Phi was discovered by Greek mathematicians, never ones to leave such a nicely self-fulfilling number alone once they’d got hold of it, and rapidly realised something quite nice concerning phi and rectangles. If you take a rectangle with a short side of length 1 unit and a long side of length Φ units, and then cut away from that a square with side length 1 unit, then the little rectangle you get left over will be the same shape as your original rectangle; the ratio of its side lengths is 1:1.618. It also just so happens that a rectangle this shape looks very… balanced and aesthetically pleasing, and so our overenthusiastic Greek mathematician friends dubbed this shape ‘the perfect rectangle’ and called phi ‘the golden ratio’.

Phi found its way back into the mathematical world several hundred years later in the early 13th century when a Pisa-born (Pisan? Pisaish? Not sure) mathematician called Leonardo Fibonacci started messing about with what would later become one of the most famous mathematical sequences of all time. The Fibonacci sequence is a very simple business; start with two ones and then, for each successive term, add the previous two. So we start with 1 + 1 = 2, then 1 + 2 = 3, then 2 + 3 = 5, then 8, 13, 21, 34 and so on. The reason it has a relation to phi is that if you divide two successive terms of the sequence by one another then you get an approximation to phi, with the approximation getting more accurate as you go further up the sequence. It starts off rather vague (1/1=1 and 2/1=2 aren’t even close), but before long things start to converge (8/5=1.6, much more like it), until eventually we arrive at something very very close (610/377= 1.618037, accurate to five significant figures). This, once again has a geometrical analogy; if you stick two squares of side length 1 unit together, and then add a square of side length two units, and then one of side length three and so on, you start building up an increasingly large rectangle; a rectangle, moreover, that starts to look suspiciously like our old friend ‘the perfect rectangle’ the more squares we add.

However, the reason phi has got so many people worked up and excited over the years is its habit of turning up in nature; although, it must be said, it doesn’t do so nearly as often as people think. A good example occurs in flowers; if you count the petals on flowers, the final number is often one of those in the Fibonacci sequence (so you get three-leafed clovers one hell of a lot more than four leaved clovers). One flower of particular interest is the rose, which often has eight on the inside and five around the outside to make 13 overall; 3 Fibonacci numbers. There are even arguments that pineapple skins and sunflowers share this feature, but trying to explain that without pictures is rather beyond my capabilities. Nobody’s entirely sure why this is, but many attribute it to a mixture of luck and confirmation bias; once somebody tells you about phi, it’s hard to stop seeing it everywhere and to ignore the countless occasions when it doesn’t crop up. I mean, 3, 5 and 8 are hardly uncommon numbers off their own bat.

However, this hasn’t deterred supporters of the theory, who claim phi turns up literally everywhere; far more often than it actually does, in fact. There are three commonly stated examples of complete phi-related bullshit that are particularly aggravating to those who know about them. The first concerns the Parthenon, in Athens, of which it is said that if you look at it front on the shape of its profile fits exactly into a perfect rectangle. Even if it did, this wouldn’t be too surprising, for as we’ve said the perfect rectangle happens to be an inherently aesthetically pleasing shape that it would not be too surprising to see incorporated into architecture to make a building look good, but the fact is that this claim is totally wrong. Pictures claiming to show it always leave out a few stairs at the bottom, or use a slightly imperfect rectangle; the relationship is close, but not ‘perfect’ as some people like to believe.

The Da Vinci connection to phi is, perhaps surprisingly, not confined just to Dan Brown; after Fibonacci, Da Vinci’s tutor Luca Pacioli was the first person to write about it (his book was entitled ‘the divine proportion’, Φ’s other nickname), and did so in a book that Da Vinci apparently illustrated. He definitely knew about the thing, therefore, but didn’t use it to compose either the Mona Lisa or the Vitruvian man. In fact, the name of the latter work gives a clue as to where its dimensions come from; Vitruvius was a Roman now known as ‘the world’s first engineer’, who used proportions of the ‘ideal’ human body (or at least what the Romans thought of it) when designing buildings. His dimensions, however, were based merely on the idea that one’s armspan and height are equal and eight times the height of the head, and didn’t use phi at all. Many phi supporters will tell you that phi does crop up a lot when measuring the human body, and in some people it does; but if we look at anthropometric data to get average data, the number of times phi appears drops markedly. In any case; there is a LOT to measure in the human body, and frankly it would be more surprising if a few of the ratios didn’t end up being phi, particularly what with it being a ratio our eye has evolved to find pleasing.

And then there’s the nautilus; an incredibly beautiful deep-sea mollusc that spends its days bobbing up quite happily in its remarkable spiral-shaped shell. However, some will tell you that such a shell is, in fact a ‘golden spiral’,  getting further away from its centre point by a factor of Φ every quarter-turn (this is the typical way of measuring spirals, because REASONS). Unfortunately, this theory was shot down in 1999 when an American mathematician named Clement Falbo decided that the best way to spend his time was to measure a few hundred shells and work out an average. His results came to an average spiral ratio of 1.33:1, making the nautilus the bearer of just another old-fashioned logarithmic spiral (incidentally, there are other, far less pretty, molluscs that do have ‘golden shells’, but people tend to forget about them for some reason).

The ‘golden ratio’ is an interesting little piece of mathematics, the kind of thing that nerds make jokes about on the internet and inconceivably bored teenagers mess around with on calculators at the back of Friday afternoon geography (I speak from extensive personal experience). It pops up in a lot of places and has several interesting properties; but some divine mathematical instrument with which to describe the whole natural world?

…might be going a bit far.