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.

An Opera Posessed

My last post left the story of JRR Tolkein immediately after his writing of his first bestseller; the rather charming, lighthearted, almost fairy story of a tale that was The Hobbit. This was a major success, and not just among the ‘children aged between 6 and 12’ demographic identified by young Rayner Unwin; adults lapped up Tolkein’s work too, and his publishers Allen & Unwin were positively rubbing their hands in glee. Naturally, they requested a sequel, a request to which Tolkein’s attitude appears to have been along the lines of ‘challenge accepted’.

Even holding down the rigours of another job, and even accounting for the phenomenal length of his finished product, the writing of a book is a process that takes a few months for a professional writer (Dame Barbara Cartland once released 25 books in the space of a year, but that’s another story), and perhaps a year or two for an amateur like Tolkein. He started writing the book in December 1937, and it was finally published 18 years later in 1955.

This was partly a reflection of the difficulties Tolkein had in publishing his work (more on that later), but this also reflects the measured, meticulous and very serious approach Tolkein took to his writing. He started his story from scratch, each time going in a completely different direction with an entirely different plot, at least three times. His first effort, for instance, was due to chronicle another adventure of his protagonist Bilbo from The Hobbit, making it a direct sequel in both a literal and spiritual sense. However, he then remembered about the ring Bilbo found beneath the mountains, won (or stolen, depending on your point of view) from the creature Gollum, and the strange power it held; not just invisibility, as was Bilbo’s main use for it, but the hypnotic effect it had on Gollum (he even subsequently rewrote that scene for The Hobbit‘s second edition to emphasise that effect). He decided that the strange power of the ring was a more natural direction to follow, and so he wrote about that instead.

Progress was slow. Tolkein went months at a time without working on the book, making only occasional, sporadic yet highly focused bouts of progress. Huge amounts were cross-referenced or borrowed from his earlier writings concerning the mythology, history & background of Middle Earth, Tolkein constantly trying to make his mythic world feel and, in a sense, be as real as possible, but it was mainly due to the influence of his son Christopher, who Tolkein would send chapters to whilst he was away fighting the Second World War in his father’s native South Africa, that the book ever got finished at all. When it eventually did, Tolkein had been working the story of Bilbo’s son Frodo and his adventure to destroy the Ring of Power for over 12 years. His final work was over 1000 pages long, spread across six ‘books’, as well as being laden with appendices to explain & offer background information, and he called it The Lord of The Rings (in reference to his overarching antagonist, the Dark Lord Sauron).

A similar story had, incidentally, been attempted once before; Der Ring des Nibelungen is an opera (well, four operas) written by German composer Richard Wagner during the 19th century, traditionally performed over the course of four consecutive nights (yeah, you have to be pretty committed to sit through all of that) and also known as ‘The Ring Cycle’- it’s where ‘Ride of The Valkyries’ comes from. The opera follows the story of a ring, made from the traditionally evil Rhinegold (gold panned from the Rhine river), and the trail of death, chaos and destruction it leaves in its wake between its forging & destruction. Many commentators have pointed out the close similarities between the two, and as a keen follower of Germanic mythology Tolkein certainly knew the story, but Tolkein rubbished any suggestion that he had borrowed from it, saying “Both rings were round, and there the resemblance ceases”. You can probably work out my approximate personal opinion from the title of this post, although I wouldn’t read too much into it.

Even once his epic was finished, the problems weren’t over. Once finished, he quarrelled with Allen & Unwin over his desire to release LOTR in one volume, along with his still-incomplete Silmarillion (that he wasn’t allowed to may explain all the appendices). He then turned to Collins, but they claimed his book was in urgent need of an editor and a license to cut (my words, not theirs, I should add). Many other people have voiced this complaint since, but Tolkein refused and ordered Collins to publish by 1952. This they failed to do, so Tolkein wrote back to Allen & Unwin and eventually agreed to publish his book in three parts; The Fellowship of The Ring, The Two Towers, and The Return of The King (a title Tolkein, incidentally, detested because it told you how the book ended).

Still, the book was out now, and the critics… weren’t that enthusiastic. Well, some of them were, certainly, but the book has always had its detractors among the world of literature, and that was most certainly the case during its inception. The New York Times criticised Tolkein’s academic approach, saying he had “formulated a high-minded belief in the importance of his mission as a literary preservationist, which turns out to be death to literature itself”, whilst others claimed it, and its characters in particular, lacked depth. Even Hugo Dyson, one of Tolkein’s close friends and a member of his own literary group, spent public readings of the book lying on a sofa shouting complaints along the lines of “Oh God, not another elf!”. Unlike The Hobbit, which had been a light-hearted children’s story in many ways, The Lord of The Rings was darker & more grown up, dealing with themes of death, power and evil and written in a far more adult style; this could be said to have exposed it to more serious critics and a harder gaze than its predecessor, causing some to be put off by it (a problem that wasn’t helped by the sheer size of the thing).

However, I personally am part of the other crowd, those who have voiced their opinions in nearly 500 five-star reviews on Amazon (although one should never read too much into such figures) and who agree with the likes of CS  Lewis, The Sunday Telegraph and Sunday Times of the time that “Here is a book that will break your heart”, that it is “among the greatest works of imaginative fiction of the twentieth century” and that “the English-speaking world is divided into those who have read The Lord of the Rings and The Hobbit and those who are going to read them”. These are the people who have shown the truth in the review of the New York Herald Tribune: that Tolkein’s masterpiece was and is “destined to outlast our time”.

But… what exactly is it that makes Tolkein’s epic so special, such a fixture; why, even years after its publication as the first genuinely great work of fantasy, it is still widely regarded as the finest work the genre has ever produced? I could probably write an entire book just to try and answer that question (and several people probably have done), but to me it was because Tolkein understood, absolutely perfectly and fundamentally, exactly what he was trying to write. Many modern fantasy novels try to be uber-fantastical, or try to base themselves around an idea or a concept, in some way trying to find their own level of reality on which their world can exist, and they often find themselves in a sort of awkward middle ground, but Tolkein never suffered that problem because he knew that, quite simply, he was writing a myth, and he knew exactly how that was done. Terry Pratchett may have mastered comedic fantasy, George RR Martin may be the king of political-style fantasy, but only JRR Tolkein has, in recent times, been able to harness the awesome power of the first source of story; the legend, told around the campfire, of the hero and the villain, of the character defined by their virtues over their flaws, of the purest, rawest adventure in the pursuit of saving what is good and true in this world. These are the stories written to outlast the generations, and Tolkein’s mastery of them is, to me, the secret to his masterpiece.