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.

The Cross

Humankind has long been inventive when it comes to the sphere of killing one another; I could probably write a whole other blog solely on the subject of weaponry for the next 50 years before running low on material, and that doesn’t even approach the field of organised execution. Hanging and stoning are two old-as-the-hills methods still, unfortunately, in use in some parts of the world, and countless others have been developed with varying degrees of complexity, pain and success involved. However, one execution method has proved to carry more cultural weight than all others, and mostly thanks to one man; I speak, of course, of crucifixion.

We all think of crucifixion as a Roman punishment, but like so many Roman things it wasn’t their invention (seriously, even their religion was nicked from the Greeks). Crucifixion first started off in Persia in around the 6th century BC, in the area that would later become the Seleucid Empire after Alexander the Great went and conquered all of it. Like so many other things, the practice later spread across the remnants of Alexander’s Empire, including his native Greece, and here it began making its way towards the ‘civilised’ world of the time. The Greeks were, apparently, generally opposed to this horrible method of execution and used it very sparingly, but much of Alexander’s old Empire would later find its way into Roman hands, and so the idea eventually made its way to Rome. Given that this was a culture whose primary form of entertainment (garnering hundreds of thousands of spectators, something even modern sporting culture can’t match) involved various people and animals dressing up to kill one another in as ‘entertaining’ a fashion as possible, it is perhaps not surprising that the Romans thought crucifixion showed potential as an execution method, particularly for those they wanted to make an example of.

This is hardly surprising; of all humanity’s execution methods, few can rival crucifixion when it comes to being horrifying and showy. This is partly helped, slightly bizarrely, by its cheapness; to show them off to the general populace, something like hanging or beheading would require some sort of raised platform, which covers only a small area and takes a decent amount of time and energy to create. The Roman alternative (the arena) was even more expensive, requiring an investment in either animals or an elaborate set of costumes and procedure in order to provide an ‘entertaining’ execution, and given that games were generally free to go and watch (paid for by the emperor or local governor to curry goodwill with the populace) it wasn’t going to pay itself back. By contrast, the sum total of all monetary investment required for crucifixion is two long sticks, some rope or nails, and a bloke to affix the resulting structure to; the crosses were even moved to the required site by the prisoners themselves, and erecting them took a few soldiers almost no time at all. This cheapness made it easy to show off their victims on a vast scale; after the gladiator Spartacus’ slave revolt was crushed in 71BC, the 6,000 captured prisoners were all crucified along the Appian way, a trail of crosses stretching from Rome to Capua. That’s 200 kilometres (125 miles), along both sides of the road. A forceful example indeed.

The very nature of crucifixion itself also helps when it comes to being showy. The crosses used in crucifixion were big old things, three or four metres tall if they’re an inch, just to ensure the unfortunate victim could be seen from great distances away. The mechanics of the execution build on this; it is often assumed that death by crucifixion comes from exhaustion, hunger, pain and blood loss, but in fact crucifixion causes death by suffocation as much as anything. With one’s upper body held only by spread eagled arms, it becomes very tiring to keep it in position, and one’s head and torso tend to fall forwards after time. However, with the arms pinned in position this stretches out one’s joints extremely painfully, offering no respite from the agony, and pulls upwards on the ribcage. This in turn puts extreme stress on the diaphragm, meaning it has to pull one’s entire weight upward every time you attempt to take a breath, and crushes the lungs under one’s own weight, slowly squeezing the air and life out of the victim. If the executors were feeling kind, then the victim would be tied to the cross, resulting in a slower but slightly less agonisingly painful death. However, Jesus was famously attached to his cross by nails through his feet and wrists (some versions say the hands, but the flesh there isn’t strong enough to hold up the weight of a body properly), and whilst this could offer the possibility of blessedly quick unconsciousness and death due to blood loss and the extreme pain, the sheer agony of the experience doesn’t bear thinking about. No matter how devoted to their cause the victim was, their screams must have undoubtedly echoed for miles as they died, just adding to the showiness of their death. Crucifixion was the ultimate tool, for the Romans, for sending out a warning, a very obvious, demonstrative way of discouraging people from following the lead of the victim.

That this approach failed somewhat is like saying the Pope thinks God is a kinda alright guy; crucifixion has guaranteed martyrdom for countless early saints and, of course, Jesus. The concept of ‘he suffered and died on the cross for us’ is, more than anything, the fundamental message of Christianity, embodying the idea of undergoing extreme pain and hardship simply to try and do right by the world and emphasising the pure and unadulterated goodness of Jesus as a person. But this has had an unexpected effect in the long run; since the story is told so often to children, the gory details are often glossed over, or the story simply because so fundamental and oft-told that it becomes very easy to forget just how horrific his agony would have been. Even this post has treated the subject of crucifixion with a decidedly neutral tone, without considering properly just how horrible it is to inflict this level of pain onto a fellow human being. Crucifixion might have been abolished by the Roman Empire 1600 years ago (by Emperor Constatine, if you’re wondering), but it would not do to forget it. Very few things are ever worth forgetting, and torture and murder are most certainly not among them.

The Conquest of Air

Everybody in the USA, and in fact just about everyone across the world, has heard of Orville and Wilbur Wright. Two of the pioneers of aviation, when their experimental biplane Flyer achieved the first ever manned, powered, heavier-than-air flight on the morning of December 17, 1903, they had finally achieved one of man’s long-held dreams; control and mastery of air travel.

However, what is often puzzling when considering the Wright brothers’ story is the number of misconceptions surrounding them. Many, for instance, are under the impression that they were the first people to fly at all, inventing all the various technicalities of lift, aerofoil structures and control that are now commonplace in today’s aircraft. In fact, the story of flight, perhaps the oldest and maddest of human ambitions, an idea inspired by every time someone has looked up in wonder at the graceful flight of a bird, is a good deal older than either of them.

Our story begins, as does nearly all technological innovation, in imperial China, around 300 BC (the Greek scholar Archytas had admittedly made a model wooden pigeon ‘fly’ some 100 years previously, but nobody is sure exactly how he managed it). The Chinese’s first contribution was the invention of the kite, an innovation that would be insignificant if it wasn’t for whichever nutter decided to build one big enough to fly in. However, being strapped inside a giant kite and sent hurtling skywards not only took some balls, but was heavily dependent on wind conditions, heinously dangerous and dubiously useful, so in the end the Chinese gave up on manned flight and turned instead to unmanned ballooning, which they used for both military signalling and ceremonial purposes. It isn’t actually known if they ever successfully put a man into the air using a kite, but they almost certainly gave it a go. The Chinese did have one further attempt, this time at inventing the rocket engine, some years later, in which a young and presumably mental man theorised that if you strapped enough fireworks to a chair then they would send the chair and its occupants hurtling into the night sky. His prototype (predictably) exploded, and it wasn’t for two millennia, after the passage of classical civilisation, the Dark Ages and the Renaissance, that anyone tried flight again.

That is not to say that the idea didn’t stick around. The science was, admittedly beyond most people, but as early as 1500 Leonardo da Vinci, after close examination of bird wings, had successfully deduced the principle of lift and made several sketches showing designs for a manned glider. The design was never tested, and not fully rediscovered for many hundreds of years after his death (Da Vinci was not only a controversial figure and far ahead of his time, but wrote his notebooks in a code that it took centuries to decipher), but modern-day experiments have shown that his design would probably have worked. Da Vinci also put forward the popular idea of ornithopters, aircraft powered by flapping motion as in bird wings, and many subsequent attempts at flight attempted to emulate this method of motion. Needless to say, these all failed (not least because very few of the inventors concerned actually understood aerodynamics).

In fact, it wasn’t until the late 18th century that anyone started to really make any headway in the pursuit of flight. In 1783, a Parisian physics professor, Jacques Charles, built on the work of several Englishmen concerning the newly discovered hydrogen gas and the properties and behaviour of gases themselves. Theorising that, since hydrogen was less dense than air, it should follow Archimedes’ principle of buoyancy and rise, thus enabling it to lift a balloon, he launched the world’s first hydrogen balloon from the Champs du Mars on August 27th. The balloon was only small, and there were significant difficulties encountered in building it, but in the design process Charles, aided by his engineers the Roberts brothers, invented a method of treating silk to make it airtight, spelling the way for future pioneers of aviation. Whilst Charles made some significant headway in the launch of ever-larger hydrogen balloons, he was beaten to the next significant milestones by the Montgolfier brothers, Joseph-Michel and Jacques-Etienne. In that same year, their far simpler hot-air balloon designs not only put the first living things (a sheep, rooster and duck) into the atmosphere, but, just a month later, a human too- Jacques-Etienne was the first European, and probably the first human, ever to fly.

After that, balloon technology took off rapidly (no pun intended). The French rapidly became masters of the air, being the first to cross the English Channel and creators of the first steerable and powered balloon flights. Finally settling on Charles’ hydrogen balloons as a preferable method of flight, blimps and airships began, over the next century or so, to become an accepted method of travel, and would remain so right up until the Hindenburg disaster of 1937, which rather put people off the idea. For some scientists and engineers, humankind had made it- we could now fly, could control where we were going at least partially independent of the elements, and any attempt to do so with a heavier-than-air machine was both a waste of time and money, the preserve of dreamers. Nonetheless, to change the world, you sometimes have to dream big, and that was where Sir George Cayley came in.

Cayley was an aristocratic Yorkshireman, a skilled engineer and inventor, and a magnanimous, generous man- he offered all of his inventions for the public good and expected no payment for them. He dabbled in a number of fields, including seatbelts, lifeboats, caterpillar tracks, prosthetics, ballistics and railway signalling. In his development of flight, he even reinvented the wheel- he developed the idea of holding a wheel in place using thin metal spokes under tension rather than solid ones under compression, in an effort to make the wheels lighter, and is thus responsible for making all modern bicycles practical to use. However, he is most famous for being the first man ever, in 1853, to put somebody into the air using a heavier-than-air glider (although Cayley may have put a ten-year old in a biplane four years earlier).

The man in question was Cayley’s chauffeur (or butler- historical sources differ widely), who was (perhaps understandably) so hesitant to go in his boss’ mental contraption that he handed in his notice upon landing after his flight across Brompton Dale, stating  as his reason that ‘I was hired to drive, not fly’. Nonetheless, Cayley had shown that the impossible could be done- man could fly using just wings and wheels. He had also designed the aerofoil from scratch, identified the forces of thrust, lift, weight and drag that control an aircraft’s movements, and paved the way for the true pioneer of ‘heavy’ flight- Otto Lilienthal.

Lilienthal (aka ‘The Glider King’) was another engineer, making 25 patents in his life, including a revolutionary new engine design. But his fame comes from a world without engines- the world of the sky, with which he was obsessed. He was just a boy when he first strapped wings to his arms in an effort to fly (which obviously failed completely), and later published works detailing the physics of bird flight. It wasn’t until 1891, aged 43, once his career and financial position was stable and he had finished fighting in the Franco-Prussian War, that he began to fly in earnest, building around 12 gliders over a 5-year period (of which 6 still survive). It might have taken him a while, but once he started there was no stopping him, as he made over 2000 flights in just 5 years (averaging more than one every day). During this time he was only able to rack up 5 hours of flight time (meaning his average flight time was just 9 seconds), but his contribution to his field was enormous. He was the first to be able to control and manoeuvre his machines by varying his position and weight distribution, a factor whose importance he realised was absolutely paramount, and also recognised that a proper understanding of how to achieve powered flight (a pursuit that had been proceeding largely unsuccessfully for the past 50 years) could not be achieved without a basis in unpowered glider flight, in recognising that one must work in harmony with aerodynamic forces. Tragically, one of Lilienthal’s gliders crashed in 1896, and he died after two days in hospital. But his work lived on, and the story of his exploits and his death reached across the world, including to a pair of brothers living in Dayton, Ohio, USA, by the name of Wright. Together, the Wright brothers made huge innovations- they redesigned the aerofoil more efficiently, revolutionised aircraft control using wing warping technology (another idea possibly invented by da Vinci), conducted hours of testing in their own wind tunnel, built dozens of test gliders and brought together the work of Cayley, Lilienthal, da Vinci and a host of other, mostly sadly dead, pioneers of the air.  The Wright brothers are undoubtedly the conquerors of the air, being the first to show that man need not be constrained by either gravity or wind, but can use the air as a medium of travel unlike any other. But the credit is not theirs- it is a credit shared between all those who have lived and died in pursuit of the dream of fling like birds. To quote Lilienthal’s dying words, as he lay crippled by mortal injuries from his crash, ‘Sacrifices must be made’.