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.

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Hitting the hay

OK, so it was history last time, so I’m feeling like a bit of science today. So, here is your random question for today; are the ‘leaps of faith’ in the Assassin’s Creed games survivable?

Between them, the characters of Altair, Ezio and Connor* jump off a wide variety of famous buildings and monuments across the five current games, but the jump that springs most readily to mind is Ezio’s leap from the Campanile di San Marco, in St Mark’s Square, Venice, at the end of Assassin’s Creed II. It’s not the highest jump made, but it is one of the most interesting and it occurs as part of the main story campaign, meaning everyone who’s played the game through will have made the jump and it has some significance attached to it. It’s also a well-known building with plenty of information on it.

[*Interesting fact; apparently, both Altair and Ezio translate as ‘Eagle’ in some form in English, as does Connor’s Mohawk name (Ratonhnhaké;ton, according to Wikipedia) and the name of his ship, the Aquila. Connor itself translates as ‘lover of wolves’ from the original Gaelic]

The Campanile as it stands today is not the same one as in Ezio’s day; in 1902 the original building collapsed and took ten years to rebuild. However, the new Campanile was made to be cosmetically (if not quite structurally) identical to the original, so current data should still be accurate. Wikipedia again tells me the brick shaft making up the bulk of the structure accounts for (apparently only) 50m of the tower’s 98.6m total height, with Ezio’s leap (made from the belfry just above) coming in at around 55m. With this information we can calculate Ezio’s total gravitational potential energy lost during his fall; GPE lost = mgΔh, and presuming a 70kg bloke this comes to GPE lost= 33730J (Δ is, by the way, the mathematical way of expressing a change in something- in this case, Δh represents a change in height). If his fall were made with no air resistance, then all this GPE would be converted to kinetic energy, where KE = mv²/2. Solving to make v (his velocity upon hitting the ground) the subject gives v = sqrt(2*KE/m), and replacing KE with our value of the GPE lost, we get v = 31.04m/s. This tells us two things; firstly that the fall should take Ezio at least three seconds, and secondly that, without air resistance, he’d be in rather a lot of trouble.

But, we must of course factor air resistance into our calculations, but to do so to begin with we must make another assumption; that Ezio reaches terminal velocity before reaching the ground. Whether this statement is valid or not we will find out later. The terminal velocity is just a rearranged form of the drag equation: Vt=sqrt(2mg/pACd), where m= Ezio’s mass (70kg, as presumed earlier), g= gravitational field strength (on Earth, 9.8m/s²), p= air density (on a warm Venetian evening at around 15 degrees Celcius, this comes out as 1.225kg/m³), A= the cross-sectional area of Ezio’s falling body (call it 0.85m², presuming he’s around the same size as me) and Cd= his body’s drag coefficient (a number evaluating how well the air flows around his body and clothing, for which I shall pick 1 at complete random). Plugging these numbers into the equation gives a terminal velocity of 36.30m/s, which is an annoying number; because it’s larger than our previous velocity value, calculated without air resistance, of 31.04m/s, this means that Ezio definitely won’t have reached terminal velocity by the time he reaches the bottom of the Campanile, so we’re going to have to look elsewhere for our numbers. Interestingly, the terminal velocity for a falling skydiver, without parachute, is apparently around 54m/s, suggesting that I’ve got numbers that are in roughly the correct ballpark but that could do with some improvement (this is probably thanks to my chosen Cd value; 1 is a very high value, selected to give Ezio the best possible chance of survival, but ho hum)

Here, I could attempt to derive an equation for how velocity varies with distance travelled, but such things are complicated, time consuming and do not translate well into being typed out. Instead, I am going to take on blind faith a statement attached to my ‘falling skydiver’ number quoted above; that it takes about 3 seconds to achieve half the skydiver’s terminal velocity. We said that Ezio’s fall from the Campanile would take him at least three seconds (just trust me on that one), and in fact it would probably be closer to four, but no matter; let’s just presume he has jumped off some unidentified building such that it takes him precisely three seconds to hit the ground, at which point his velocity will be taken as 27m/s.

Except he won’t hit the ground; assuming he hits his target anyway. The Assassin’s Creed universe is literally littered with indiscriminate piles/carts of hay and flower petals that have been conveniently left around for no obvious reason, and when performing a leap of faith our protagonist’s always aim for them (the AC wiki tells me that these were in fact programmed into the memories that the games consist of in order to aid navigation, but this doesn’t matter). Let us presume that the hay is 1m deep where Ezio lands, and that the whole hay-and-cart structure is entirely successful in its task, in that it manages to reduce Ezio’s velocity from 27m/s to nought across this 1m distance, without any energy being lost through the hard floor (highly unlikely, but let’s be generous). At 27m/s, the 70kg Ezio has a momentum of 1890kgm/s, all of which must be dissipated through the hay across this 1m distance. This means an impulse of 1890Ns, and thus a force, will act upon him; Impulse=Force x ΔTime. This force will cause him to decelerate. If this deceleration is uniform (it wouldn’t be in real life, but modelling this is tricky business and it will do as an approximation), then his average velocity during his ‘slowing’ period will come to be 13.5m/s, and that this deceleration will take 0.074s. Given that we now know the impulse acting on Ezio and the time for which it acts, we can now work out the force upon him; 1890 / 0.074 = 1890 x 13.5 = 26460N. This corresponds to 364.5m/s² deceleration, or around 37g’s to put it in G-force terms. Given that 5g’s has been known to break bones in stunt aircraft, I think it’s safe to say that quite a lot more hay, Ezio’s not getting up any time soon. So remember; next time you’re thinking of jumping off a tall building, I would recommend a parachute over a haystack.

N.B.: The resulting deceleration calculated in the last bit seems a bit massive, suggesting I may have gone wrong somewhere, so if anyone has any better ideas of numbers/equations then feel free to leave them below. I feel here is also an appropriate place to mention a story I once heard concerning an air hostess whose plane blew up. She was thrown free, landed in a tree on the way down… and survived.

EDIT: Since writing this post, this has come into existence, more accurately calculating the drag and final velocity acting on the falling Assassin. They’re more advanced than me, but their conclusion is the same; I like being proved right :).