Sunday, 29 April 2012

The Physics of Soccer: How does 'Curl' work?

Welcome to the second post in our series on the physics of sport. How can soccer players cause a ball's trajectory to change in the air? And what are the different ways they can do this?

Last week we posted on how 'swing bowling' worked in cricket. Ian Lynch (@Ingotian on twitter) pointed out that the physics would surely be very similar in each of the subjects, indeed the end results are practically identical. However, when we investigate further, we realise that there are some distinct differences between the two. Before we get to those, lets watch a video of Roberto Carlos putting this skill in action:
This is one of the most extreme exhibitions of curl ever shown, and shows the effect easily. You can see, especially from the angle at 52 seconds, that ball originally does not go towards the goal. However, after a very short time, it seems to turn strongly and goes in. So what is it that leads to this strange phenomenon?

If you watch closely, you will notice that Carlos hit the ball slightly off centre. This gives the ball spin, and it is this spin that leads to the curl. The reason for this is due to something called the Magnus effect. This principle says that when an object spins in a fluid, whether a gas or a liquid, it creates a sort of whirlpool around itself, which leads to the object being pulled to one side. When a ball moves without spin, the air around it looks like this:

As air passes over the curved surfaces, it is forced to speed up. When a fast-moving fluid passes over an object, it puts less pressure on it than if the fluid was moving slowly. However, because all of the ball's surface is curved, the effects cancel out and the ball travels straight. However, the effect is different when the ball is spinning:

The side on the top is spinning with the airflow. This creates less drag on the air on that side and therefore it moves faster. Because the air is moving faster, it has lower pressure. The ball feels a force towards the patch of low pressure, causing it to curve towards that way.

The force can be put in any direction. A ball that has top spin, which is more applicable to a sport like tennis, will dip very quickly, making it very easy to hit hard and still get in the court, whereas a ball with backspin will instead glide as the Magnus effect will help to give it lift.

The Magnus effect is a more specific part of Bernoulli's principle, which says that if a fluid's speed increases, its pressure decreases. This is used extensively in engineering. For example, this is a diagram of an airplane wing:

The air on the top curved side has to move faster, lowering its pressure, and causing the plane to move up. We can also see this in other sports, like golf.

There is another force that increases this effect, called the 'Wake deflection force'. Last week, we talked about the boundary layer of a travelling ball:

This is where we start to see the similarities between cricket ball swing, and soccer curl. Just as the smooth and rough sides of the cricket ball would cause the air going past it to go in a specific direction, so does the spin of the soccer ball:

The famous quotation from Newton, says that 'Every action has an equal and opposite reaction'. Therefore, the action of the Wake being pushed downwards has the reaction of pushing the ball upwards, creating a force which pushes in the same direction as the force from the Magnus effect, increasing the overall movement. However, this Wake deflection force does not apply at very fast speeds, when the air cannot be deflected. This means that on very powerful strikes, the ball travels without curving for a short amount of time before it slows down enough that the air can catch it and it can start to curve. This amount of time is usually very short, but this is what leads to the unpredictability of well hit curling shots.

And my last point on the subject of curl, as an Englishman, I am obliged to show what is perhaps one of the greatest curling free kicks of all time:

Ok, so we have finished talking about curl, haven't we? We learnt about spin, but have a look at this video:

From the angle at 5 seconds, we can easily see the ball curve in the air. However if we watch the ball from the angle at 10 seconds, it spins, but not enough to create the large curve in its trajectory that we see. This type of free kick is known as Cristiano Ronaldo's 'Knuckleball', and there is a very specific technique to it. Ronaldo hits the ball very differently to when he wants to do a normal curling shot. Instead of hitting with the inside of his foot, he hits right in the centre of the ball with the laces of his boot. This means he can put a huge amount of power into the ball without making it spin.

In the last part of the post, we considered the ball as being perfectly smooth. However, in practice, it is not. The ball has prominent seams. When the ball is spinning fast, the change from these seams is unnoticeable. But when the ball is barely spinning, like in the Knuckleball, they can catch the air. The ball floats, and then dips quickly. What makes this even more devastating, is that the air can catch on one side but not the other very easily. This means that the ball acts almost like a beach ball. It glides through the air, and can move in any direction with incredible ease. The changes can come very late in the ball's flight and all of these effects add up to create an almost impossible to predict trajectory. This sort of thing, though not seen often in soccer, can be very common in other sports. For example, the name 'Knuckleball' came from a pitch in baseball which employed the same properties to get the same results. These pitches/shots are very effective, but the skill to hit or throw a ball and put no spin on it is one that requires huge amounts of practice and a large amount of skill. Cristiano Ronaldo, one of the best soccer players in the world, is one of few who have perfected it.

To conclude, Spin causes a ball to curl. The faster the spin, the more extremely the ball curls. However, due to minor imperfections in the ball, if a player can hit it without spin, it will still curl in a chaotic manner that can create big problems for goalkeepers.

We hope you enjoyed this post. If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at or search "The Aftermatter"on Facebook.

Ned Summers

Check out our last two posts:
The Physics of Cricket: What is 'Swing Bowling'? - The first instalment of our "The physics of sport" series. How does a bowler make a cricket ball swing? We assure you, it is very different to how soccer balls curl!
Untitled - [yes, that's the title] (Guest post from TheCompBlog) - A guest post not about physics, but just as interesting!

What are we posting about next:
The Physics of Rugby: Forward Passes and Relativity - How can a backwards pass look like they are going forward, and why is it so difficult for referees to judge?

If you have ideas for posts we would love to heard them. Contact details are above.

Sunday, 22 April 2012

The Physics of Cricket: What is 'Swing Bowling'?

Hello, and welcome to the Aftermatter for the first instalment of a series on the physics of sport in the lead up to the Olympic Games, this week talking about Swing Bowling in cricket. Any of you who have watched cricket will have seen this effect; fast bowlers bowling balls that seem to swerve in mid-air. But why does this happen? And why do they constantly rub the ball against their trousers? And what about the mysterious 'reverse' swing?

To start off with, maybe I should show a video of some excellent swing bowling, so that those of you who don't follow cricket can see the effects of the physics I'm about to explain.

Ali's bowling there was an example of conventional inswing, that is, swinging towards the right-handed batsman. But how exactly did Ali make this delivery swing?

First let's look at a new cricket ball. It is spherical, with a leather outer layer, and has a "seam" forming a circumference around it, which sticks out a bit.
A new cricket ball. Note the prominent seam
To bowl this type of swing, a bowler must release the ball with the seam pointing in the direction he wants it to swing in, and at least one side of the ball should be as smooth as possible. This is why this type of swing works best with a brand new cricket ball: neither side will be particularly rough. 

When looking at the aerodynamics of an object, it is important to consider the "boundary layer," which is the layer of air directly adjacent to the object as it moves. There are two main types of boundary layers, laminar boundary layers, and turbulent boundary layers. In laminar layers, the air particles are quite uniform, and move over one another in layers. However, in turbulent layers, the position of the air particles is quite chaotic, and their motion more random. The following diagram, courtesy of NASA, shows this.
In conventional swing, one side is always kept as smooth as possible, to try and keep a laminar boundary layer, whilst the other side is left to become rougher naturally, as the boundary layer should be turbulent on that side. This is why cricket players rub the ball against their trousers; they are trying to keep one side of the ball as smooth as possible in order to create maximum swing. The diagram below shows a conventional swing ball.

The air hits the smooth face of the ball first, and a laminar boundary layer is formed in both directions. However, as we see at the top of the diagram, the seam 'trips' the air particles of the laminar layer, and a turbulent layer is formed. Now, turbulent layers cling to the surface of objects much more than laminar layers, which leave the object relatively early. This means that there is more air resistance near the top of the ball in the diagram above, and the bottom of the ball will travel faster. Since one side of the ball is travelling faster than another, the balls direction will curve towards the side travelling slower. This starts to happen at about 30 mph, and reaches a peak at about 70 mph, depending on the ball. After 70 mph, this effect is gradually reduced, until it comes to a complete standstill at about 90-95 mph. This is because, as the ball is bowled faster, the air becomes turbulent immediately upon impact of the ball, and thus the air is turbulent on both sides, and there is no force pushing the ball in either direction. But what happens if you bowl even faster than this?
If you did begin to bowl at 100 mph or more, then you may begin to notice reverse swing with even a new ball. The air becomes turbulent on impact with the ball, but when the air is tripped by the seam this time, the layer becomes even more turbulent! Very turbulent boundary layers actually break away from the object earlier than less turbulent ones, meaning that exactly the opposite of what happens in conventional swing happens! The ball swings the opposite direction, with no change of action on the part of the bowler. This can be very difficult to face for a batsman; top batsman can often predict by the seam direction which way the ball will swing, but with reverse swing it does exactly the opposite of what they would expect.

In practice, no bowler has ever bowled faster than 100 mph, so Reverse Swing with a new ball would be almost impossible to achieve. However, as the ball becomes older and rougher, the rough surface causes the air to become turbulent at lower speeds, so with an old ball, reverse swing can be achieved at speeds of about 80 mph, which most international fast bowlers can reach. It's for this reason that the rough side faces the batsman in reverse swinging deliveries rather than the shiny side. The devastating effect of reverse swing can be seen in the video below, skip to 3:10 for the best bit.

There is one more type of swing which isn't mentioned as often, as it is used less in top level cricket, but is perfect for amateur players. In many ways, it is the most simple type of spin. In order to bowl it, one needs to have one side smooth, and the other rough.

It uses the same principles of laminar and turbulent layers as conventional and reverse swing, but rather than use the seam to trip up the boundary layers, it uses the rough surface of one side of the ball to create turbulence
At less than 70 mph, the boundary layer over the smooth side will be laminar, and over the rough side it will be turbulent, meaning the ball will swing towards the rough side as with conventional swing. You can even try this out at home quite easily: get a tennis ball, and cover one half with duct tape, and throw it straight with the duct tape on the left, and the uncovered half on the right, and you should find that it swerves right.
However, bowling at higher speeds, more than 70 mph, like in reverse swing, the boundary layer will be turbulent on both sides of the ball. The rough side makes the air even more turbulent, meaning it detaches from the ball sooner, and the ball moves in the opposite direction to how it would at lower speeds. This happens for exactly the same reason that conventional swing switches to reverse swing at a certain critical speed.

In conclusion, we have seen that physics can explain the seemingly mysterious arts used by bowlers to try and deceive batsmen, and that "Reverse" swing is just a natural continuation of conventional swing. It is so powerful because the bowler changes nothing, it is just the physics that changes, and this creates some of the greatest contests in international cricket.

We hope you enjoyed this post. If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at or search "The Aftermatter"on Facebook.

Theo Caplan

Check out our last two posts:
Untitled - [yes, that's the title] (Guest post from TheCompBlog) - A guest post not about physics, but just as interesting!
What is φ, the 'Golden Ratio' - This ratio appears in everything from art to nature, but what is it? And why is it so special?

What are we posting about next:
The Physics of Football/Soccer: What is "Curl"? - How do soccer players manage to change the trajectory of the ball as it flies through the air?

If you have ideas for posts we would love to heard them. Contact details are above.

Monday, 16 April 2012

Untitled - [yes, that's the title] (Guest post from TheCompBlog)

Nicolas Weninger is the Author of TheCompBlog, a blog based on tech, but deviates into other issues, as seen below! He is one of our partnerships and we really enjoy what he writes about. We are happy to present a guest post which may not be what you are used to on this blog, but will hopefully make you think as much as anything we have written about, so, here it is!

Having been approached by the one and only Ned Summers, asking me to write a guest post for The Aftermatter, I immediately jumped at the offer. Getting the opportunity to write on a blog featured by Stephen Fry is great, and I intend to use it to the fullest extent through a series of shameless self-promotional links.

Fine, I am only joking, but for those readers used to meticulously researched posts and articles here, let me fill you in on my blogging style. I usually start my posts with a brief introduction, stating my current position and location where appropriate, followed by 1000 words of me going on and on about a set topic, often deviating and not keeping a fluid argument going, backed up with lots of ambiguous examples.

However, I am writing this post under slightly different circumstances. Such a post is usually written on a plane on the way to or from a holiday, but today our flight is delayed by two hours, so I accidentally drank too much coke from the lounge bar, got far too hyper, and decided to start writing now in the Zurich Airport lounge, waiting for my flight back to London.

In addition to this locational difference, I decided to do a bit of research into today's topic, one that many writers have written about before, including the Ratcritic. It's about freedom of speech in an age of the digital revolution. I had initially intended to write this for my blog, but the highly intelligent readers of The Aftermatter deserved a well-researched post, though to be honest, the extent of my research will be nothing compared to the monstrous amounts that my colleagues here do for their posts. Anyway, let's begin.

The Ratcritic started my pondering into this subject with a post detailing Yanna Richards (an aspiring singer) fiasco with her school demanding that she take down a video on her (very good) YouTube channel. From that, he wrote a very good post on how the British people were afraid of offending anyone, drawing a conclusion imploring the reader to "[not] take notice of other people being offended. Ignore them, [as] they are exactly the kind of people who screw up the whole system."

Now, I understand how he reached this conclusion, but I felt that it was not adequate for what I wanted, as with such information, I would have not met the same conclusion; however, he did bring up some important points that I would like to base my post on today.

After I read his post, I remained conscious of the topic, and shortly after, the UK government really went all 1984 on us. For those of you who do not understand the reference, 1984 was a book by George Orwell depicting a dystopian world, where everything one does is monitored. Anyway, the government proposed a plan to force telecom operators (so, BT and Virgin for example) to record and store all Internet and cellular (mobile phone) activity for two whole years. This would mean that if the police wanted to, they could easily get a warrant and search through all your data in real time! To be honest, this is not much different from what happens nowadays with sites like Facebook, Google and recently, Twitter, who will give governments permission to access the information its users post, but what scares me the most about these plans is a matter of storage, and how the telecom companies, who are lobbying against this pan by the way, are possibly going to keep this information safe from hackers "licking their lips (telegraph)," who intend to use the wealth of information on there for evil deeds.

Just think about that for a moment. Two years AND real time data available at the click of a finger. Not just Internet communications, but telephone calls as well! Scary stuff, no?

What hit me hard was the last sentence of that article. "The internet is not an open, free-for-all zone," with a link to this article, about the guy who was sentenced to 60 days in prison for tweeting something controversial about Fabrice Muamba. The Telegraph said, "Police were inundated with complaints as members of the public reported the student's comments." Does this not look like communism all over again: snitching on someone for expressing his or her own opinion? First off, let me say that the comments were not nice, especially considering the position he was in at the time, but in a country that prides itself on being liberal, freedom of religious beliefs, free to express an opinion, this sentence is abhorrent. The judge claimed that, "there is no alternative to an immediate prison sentence." Why might this be the case, I ask him? You would prefer to ruin a 21 year-olds life by giving him a criminal record rather than keeping to the very law that created the wealth in Britain?  Never mind the fact that Britain has no constitution [the unwritten constitution], which in itself is a whole other post.

Back to the quote about the Internet not being free. That's a real shame. The Internet has evolved to be so much more than a simple one-way content distribution platform, like TV, as I have repeated many times before. The Internet is one of the panicles of the 21-century, and having it censored like this will destroy a platform that has been a catalyst of world change. I would understand monitoring for potential terrorist threats (although not on the scale mentioned beforehand), but censoring to make sure that someone's seemingly insignificant opinion can never reach the world. I am growing up in a time where the potential of a global network of networks has only just started to be tapped, and I would like to see that potential become reality sooner than later, to quote a song by Matt Kearney.

What's even more disturbing is that in the US, employers are asking potential candidates for their Facebook credentials so that they can screen them and determine whether they are good enough for the job. This story blew up to the extent that the administration had to outlaw this activity, but even in UK schools, says the Ratcritic, teachers are screening profiles to check whether there is anything that could damage the school's reputation. This is almost equally as bad, as first, Facebook and the likes are your personal lives. Business and personal lives have been separate for decades, why should the case be any different now? My ICT teacher gave a lesson, to which I came along during a free games period, where he talked about the mixture of these two, and said that anyone who does not accept it will find it really hard later in life. Well, I am not accepting it! It's astounding that we should be even discussing this: before the Internet, your boss was your boss and your friend was your friend. Why then should 2012 be the year your boss becomes your friend? I say it should not. Again, you should not be compelled to censor your opinions because of a stupid mindset we have built up.

This is going on for far longer than I had planned!

It gets worse. Much worse.

Arizona House Bill 2549 makes it a crime to use any digital device to communicate using obscene, lewd or profane language or to suggest a lewd or lascivious act if done with intent to annoy, offend, harass or terrify." This would be a class 1 misdemeanor or worse! Look at the language closely, very closely. When you think about it, this bill would make it illegal to say anything bad online.

Oh, but they say it's only for cyber bullying! Good, I'm completely fine with that, but we have seen over and over again that governments abuse these laws to the extent that they threw 21 year-olds in jail for a tweet! So now we are making an ingenious platform a sterile utopia? Reality check Arizona: humans are not perfect! I refer you to my rant paragraph about how the Internet is so great, and to that I add that the Internet is a diverse place to exchange views and opinions of all sorts, like the London Royal Exchange back in the Victorian Era. You cannot possibly even attempt to censor the Royal Exchange, and no one tried to, so why should the Internet be different.

The Ratcritic concluded, "If Martin Luther King had never said that black people should be treated equally to white people, then we may still be in the same terrible position of racial discrimination." I completely agree (plus, if I don't [I do though!] the government will throw me in jail and put a criminal record on my name). MLK was unfortunate in the sense that he did not have the Internet to help him, but the same principles apply. They tried to censor him, and look what happened. We now have the ability to spread messages like MLKs at an astounding rate, and trying to censor the voice of 3 billion people will backfire to an unimaginable degree. Look at World War One: it only took one spark to ignite the war that killed 16.5 million people. It will only take one small spark, like the tweeter in jail story, or the SOPA fiasco, to release the world on a rant, much like what I did here!

We hope you liked this guest post. TheCompBlog posts about technology, but as you have seen here, there is also many posts about current affairs. We would like to hear what your opinion is on this subject and both TheCompBlog and The Aftermatter are on twitter if you would like to get in touch.

Check out our last two posts:
What is the strong force? - How does the force that holds together the most fundamental parts of all the matter in the universe work?
What is φ, the 'Golden Ratio' - This ratio appears in everything from art to nature, but what is it? And why is it so special?

If you have ideas for posts we would love to heard them. Contact details are above.

Tuesday, 10 April 2012

What is the Strong Force?

We take for granted that the world doesn't fall apart, but what is it that holds the most basic of particles together?
If we looked at any everyday object on the earth, from a glass of water, to the air, or even the computer screen you are reading this on right now we would be looking at atoms and molecules. These are held together by the electromagnetic force, but if we looked in more detail at the atoms we would realise that they are also made of smaller pieces. An atom consists of a nucleus and orbiting electrons. The nucleus consists of even smaller pieces, called protons and neutrons. However, protons all have a charge of +1, and we know that the electromagnetic force dictates that like charges repel, so how can they stay together so compactly? Well what we are seeing here is a demonstration of the strong force in action. Later in the post we will have to get into some messy particle physics, so I recommend reading our other post on the particles of the standard model.

We can see the effects of the strong force with protons, but to see where the strong force is at its most influential we have to look to the components of these protons. Protons (and neutrons) are made of quarks. Quarks are fundamental particles, they are not made of anything and cannot be broken down any further:

There are different types of quarks, called flavours. One of the things that make quarks fundamentally different to other significant particles, for example the electron, is that they are the only particles we know of  that interact through the strong force, excluding the carrier of the force, the gluon. This property leads to a somewhat strange property of quarks, they can never exist alone. They can only form in groups called hadrons (hence that large hadron collider, LHC, at CERN). These hadrons come in two varieties, baryons and mesons. Baryons consist of three quarks, or three antiquarks, whilst a meson consists of one quark and one antiquark:

So what governs which quarks attract which? Well quarks hold a sort of charge. It is intrinsically different to the charge felt by particles that interact through electromagnetism. In electromagnetism there exists only positive and negative charge, though in different values. In the strong force there are three different types of positive and negative charge and these are expressed with colours. Quarks can have either Red, Green or Blue charge and just as anti particles have the opposite electromagnetic charge to their matter counter-particles, as do the antiquarks. These charges are called Antired, Antigreen and Antiblue. I have to stress that there are no actual colours in these charges, but they are used to clasify the different types. Any normal coloured charged quark will attract any anti coloured charged quark, as opposite charges attract. For example, a red charged up quark could attract a antired, antiblue or antigreen down antiquark to form a π+ meson. Not only do the normal and anti colour charges attract, so do the different colour charges. A red quark will repel another red quark, but will attract a blue and a green. All baryons have one quark of each charge to make a RGB combination. The same happens with antiquarks, which come together in Antired-Antigreen-Antiblue triplets. All these are shown in the diagram above, with the meson having a green quark and an antigreen antiquark. Once three quarks have come together as a baryon, another of any colour charge will be attracted by two, but repelled by one. This means that the quark cannot stick to the baryon, however if that quark is part of a baryon itself, each quark in each baryon would be attracted to two quark in the other. This leads to baryons, for example protons and neutrons, being able to hold together. This is what we see in the atomic nucleus. This force is called residual strong force, or the nuclear force.

As with all the other forces, with the unusual exception of gravity, there is a particle carrier, or boson, that moves the effects of the force over distance. The electromagnetic force has the photon, and the weak force has the W and Z bosons. The strong force has a particle called the gluon. The gluon has the ability to change the colour charge of a quark. For example, if a quark has a charge R then if it emits a gluon with a sort of 'positive R, negative B' charge, then the quark will become charge B. The theory of the strong force, called quantum chromodynamics (or QCD), allows for only eight different colours of gluons. Due to the fact that gluons have colour charge, they attract each other. This property is not seen in any other bosons so instead of filling a space like a photon, the gluons concentrate in a line between the two quarks they are acting upon. In theory, this attraction could lead to gluons clustering into 'glueballs'.

This mutual attraction leads to other unusual properties and one that only the strong force has. Unlike with gravity or the electromagnetic force which weaken with distance (the hardest part of pulling two magnets from one another is the initial pull), the strong force gets stronger, like an unbreakable elastic band wrapped around the quarks. If two attracting quarks are pulled to a distance of around 10-15 meters from each other, the force required to pull them any further would be infinite. The strong force is called the strong force because, in comparison to the electromagnetic force, it is stronger, and it is also the strongest of all the forces; however, if we created energy levels similar to those soon after the big bang, we find that the weak, electromagnetic and strong force get similar strength and this point is known as the grand unification of the forces. It is a key component in forming a theory of everything.

We hope you enjoyed this post. If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at or search "The Aftermatter"on Facebook.

Ned Summers.

Check out our last two posts:
What is φ, the 'Golden Ratio' - This ratio appears in everything from art to nature, but what is it? And why is it so special?
What is thelifetime of a star? - Many of you will have learned about how a stars life unfolds, but what are the basic processes behind it all?

If you have ideas for posts we would love to heard them. Contact details are above.

Sunday, 1 April 2012

What is φ, the 'Golden Ratio'

For hundreds of years, artists, architects, musicians, mathematicians, and even biologists have been fascinated by this number, seemingly ubiquitous in all we consider beautiful. But what exactly is this number? Where is it derived from? And what gives it such strange properties, and why does it come up so often in unexpected places?

One of the simplest ways to define the golden ratio was proposed by Euclid in about 300 BC. It spoke about a particular point on a line.

Euclid investigated a special point, dividing the line a+b, into a and b, in such a way that the ratio a:b was exactly the same as a+b:a. Perhaps an easier way to envisage this is with a rectangle, often called the “Golden Rectangle.”

The ratio of the vertical side to the horizontal side of this rectangle is clearly 1:φ (pronounced "phi"). However, if we draw another vertical line to construct a square of side one, we produce another rectangle.

Euclid then defined φ as being the number for which this new rectangle formed on the right was similar to the large rectangle as a whole. More mathematically speaking,

Because this new rectangle on the right is also a golden rectangle, it itself can then be split up into a square and a golden rectangle! This process can go on ad infinitum, as shown in the diagram below.

But how do we find a value for φ? We can use the expression above
then multiply by (φ-1) to get…

-1 from both sides to get a quadratic equation

Βut since this is geometry we eliminate the negative value and are left with just 1.61803…

You may have spotted something else funny about φ. We could write our original equation slightly differently by flipping the fractions upside down.

Yes that’s right, φ has the unique and bizarre property that it is exactly one greater than its reciprocal: 1/φ = 0.61803…, but it doesn’t stop there. Using the quadratic equation we used earlier, we can get an expression for φ2...
So φ2=2.61803…! This also gives us an easy way to calculate further powers of φ, by substituting any φ2 for φ+1

There is a fundamental pattern happening here. φ=1 + 1/φ. This can also be written as

In the same way
φ2=φ+1, or φ210
In general, you could say that φn=φn-1+φn-2.
So, if we had a series, let’s call it F, of the powers of φ, then Fn=Fn-1 + Fn-2.

In other words, each term is formed by adding the two previous terms together. You may have heard of a sequence like this before, it’s called the Fibonacci series. It normally starts like this…

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 …

The only difference between this and the powers of φ is that we defined the start differently! After the first two terms it follows exactly the same rules for the formation of further terms.

So, if we consider the powers of φ being as being part of the series, then each subsequent term of the series will be φ times greater than the last. This is true by definition, for any number x xn/xn-1=x. This means that in any Fibonacci sequence, no matter what your starting terms, the ratio between one term and the next will gradually close in on φ!

Now we move on to speak about the irrationality of φ. φ is an irrational number, which means that it cannot be expressed as a fraction of two integers, and has an infinite amount of digits that continue without repetition. Another property of irrational numbers is that neither a rational multiplied by an irrational number, or a rational add an irrational number will have a rational result.

However, irrational numbers can be expressed as ‘continued fractions,’ such as the one I am about to show you.

We start off with a simple fact:

Now, wherever we see φ, we can replace it with 1+1/φ

This is actually the simplest continued fraction you can get: the 1’s gradually trailing off to infinity, but this also makes it the ‘most irrational’ number. But how can any number be ‘more’ irrational than another? How irrational a number is depends on how well it can be approximated using fractions. π, for example, can be approximated relatively accurately by 22/7, which is accurate to about  0.001264, but the closest approximation to φ with a denominator under 10 is 13/8, which is only accurate to about 0.00696601125, six times less accurate! This article explains further.

It’s for this reason that φ shows up so much in nature. For example, for most plants which grow leaves around a stem, each leaf is rotated exactly 360/φ°, or about 222.5° from the previous one. Nature is taking advantage of the irrationality of φ: this means that no two leaves will ever overlap very much, in fact it is so irrational that it creates the minimum amount of overlap between leaves. This is important because it allows more light to get at each plant so that they can photosynthesize with maximum efficiency.

Again, because it is so irrational, it can be used to aid space efficient packing of seeds. In sunflowers, the seeds are placed in two opposing spirals, one of 89 seeds, and one of 55 seeds, two Fibonacci numbers. These spirals create the Golden Angle we talked about above. This means that two seeds will never line up perfectly with the center of the sunflower, and thus there will always be a small gap for the next seed to be placed into. Other seed bearers, such as pine cones, also have patterns of two Fibonacci spirals.

One of the most famous uses of φ in nature is in the Nautilus shell, and the shells of many other creatures. There is a spiral in their shell patterns as below. This is because of the self-similarity of Golden Rectangles: the Nautilus can keep growing its shell in the same pattern and constantly get bigger, meaning the shape and design is never a limiting factor.!Fibonacci_spiral.svg
φ appears in many other natural patterns, often growth related, in other areas as well. This diagram shows the many occurrences of φ or its powers in Leonardo Da Vinci’s Vitruvian Man.

Other artists have used φ as well, such as in this piece by George Seurat (top), and this by Piet Mondrian (bottom).

Debussy, the French impressionist composer, is said to have used φ in many of his compositions, with the climax of the piece often coming approximately 61.8% of the way through the entire work, such as in Reflets dans l’Eau, where the harmonic structure also follows the Fibonacci sequence. The recursive, infinite nature of Golden Rectangles and φ seem to be very attractive to artists, so much so that it became known as the “Divine Proportion”, and perhaps that is why φ features so heavily.

Many have also used φ, with varying success, to try to predict the stock market, using techniques such as Fibonacci Retracement, or predicting population growth in various species, mostly based on a more complex version of Fibonacci’s original Rabbit Problem.

Whilst so much about this number and its ubiquity is still a mystery, one thing is certainly clear: φ has a profound effect on so much in our world, whether in nature, art, or mathematics, and we will keep discovering more appearances of this enigmatic number for years to come.

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Theo Caplan.

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