Funbers 4.6692… 5 and 6


A new addition to the list of important mathematical numbers, Feigenbaum’s constant was only discovered in 1978 through the study of chaotic systems. Chaos in the mathematical sense is pretty much what you might expect – it describes something that is completely unpredictable. My favourite way to think of it is using magnets. If you have two magnets each with a North and South pole (those funky blue and red ones from school) and hold opposite ends near to each other, then they are attracted and will quickly move together. This is a nice stable system – we know what will happen every time. And if you slightly adjust how far apart you put the magnets, or at what angle you hold them near to each other, it won’t make a difference they will still join up. Now, what happens if you try and hold two of the same ends near to each other? Well, for one they won’t attract – they repel. But what’s important here is that they move around unpredictably. Try it. If you force them towards each other they push back and can end up moving sideways, up, down, pretty much in any direction. And if you try to repeat the same movement you can’t. That’s chaos.

Below is a great example of a fractal – a repeating pattern that continues to look the same despite zooming in further and further. The rate at which the image zooms in is the Feigenbaum constant – cool, huh?


Feigenbaum’s constant is important because it describes the rate at which the simplest mathematical systems (called one-dimensional maps) descend into chaos. The really awesome thing is that you can create all kinds of different systems (following a few basic rules) and they will all go into chaos at the same rate given by this exact constant.


5 – FIVE

Apart from being a hit(?) boyband from the 90’s, five is also a number. It has the fun quirk of being the fifth number in the Fibonacci sequence: 1, 1, 2, 3, 5… where the next number is the sum of the two before it. It’s also the number of human senses (or at least the main ones) and the number of rings in the Olympic symbol. My favourite use of five, however, has to be the 5 Platonic Solids. These are the most, regular, symmetrical and beautiful shapes in all of maths – some people would say that they are so beautiful that they should be permanently inked onto your body… who are these crazy people? Hint: it’s me.

The rule is that you need a 3D solid where each face is the same shape and at each corner the same number of faces join together. Take the simplest example: the cube. We have 6 faces that are all squares and at each corner 3 squares join together. The smallest Platonic Solid is the Tetrahedron or triangle-based pyramid which has 4 faces that are all triangles. After the square comes the Octahedron which has 8 faces that are all triangles – basically two square-based pyramids stuck together so that the square face is inside the solid. The last two are the Icosahedron with 12 pentagon-shaped sides and the Dodecahedron which has 20 triangles all stuck together. These are the only 5 shapes that satisfy the very simple rule and they appear everywhere in nature from the shape of viruses to the structure of molecules. They’re great (credit to Tony Tiger).


6 – SIX

The number of impossible things that the Queen from Alice in Wonderland believes before breakfast and also the number of legs on every insect on earth. Insects are the largest group of species we have on the planet, so much so that they outnumber all of the other species combined… We also have 6 Quarks: these are some of the fundamental particles that make up our universe and they have great names too – up, down, bottom, top, charm and strange. Sounds like a fun weekend…

Mathematically, six is what we call a perfect number: all of its factors (the numbers that divide it) add up to give six: 1 + 2 + 3 = 6. It also happens to be the only number in existence where not only do all of its factors add up to give the number, they also all multiply together to give the number too: 1 x 2 x 3 = 6. Perfect numbers (the ones that equal the sum of their factors) are a little more common, the next three are: 28, 496, 8128. There are more of course, but they get a little tricky to find. Be my guest and see if you can work them out… sounds like another fun weekend to me.



The Prisoner’s Dilemma

I demonstrate the classical game theory problem live on BBC Radio, with a short introduction to the subject from Sergey Gavrilets. You can listen via the Naked Scientists here.

Sergey – In our models we included two different types of games. One is coming under the name of “Us versus Nature” games. If you think of a situation where group members have to go on a hunting trip, or you can think of a situation where you have to clear part of the forest to be able to plant some plants there.

The other type of games is us versus them games. That would be a conflict with a neighbouring group or territory over some amazing opportunities.

Tom – So would I be right in thinking the idea of us versus nature – say there are eight people in your group and you need to hunt a mammoth for food thinking hunter/gatherer style, provided four people turned up we would kill the mammoth? But then, the idea is that only  means half of the group need to give effort because the whole group will be fed, for the whole group to get the reward. Is that the kind of game theory thing where people are thinking should I commit and put the effort in or can I be lazy and free ride through – is that the idea here?

Sergey – Exactly. That the idea. Perhaps we would need more than four people to kill mammoth. Maybe more like ten but, indeed, that’s the idea. These type of games is also coming under the rubric of the volunteer’s dilemma. Just imagine a situation like you and your family are watching a movie and then there is a phone call. The phone is in the next room and you don’t want to go there but it’s so annoying. So this is a situation where everybody would benefit from somebody going and answering it but nobody wants to do it. This is a typical volunteer’s dilemma.

Tom – That ringing phone example occurs pretty much daily at my house. Sergey Gavrilets there from the University of Tennessee.

Georgia –  Now that we have a better grasp of how game theory works – Tom, I understand we’re going to do a little experiment of our own in the studio…

Tom –  Indeed we are Georgia – it’s called the prisoner’s dilemma. And as our resident mathematician I’m going to be running the show, with yourself and Peter Cowley, who’s still with us, our willing volunteers…

Georgia – How does this work?

Tom – This is a hypothetical scenario where both of you have been caught committing a crime.

Georgia – Oh no! What did we do?

Tom – Say you’ve robbed somebody – you’ve done a bit of pickpocketing. You’ve been caught by the police and the police say to you we have enough evidence to put you in jail for one year. They say this to both of you but you’re seperate and you can’t communicate. They also say to you but if you snitch on your friend and cooperate with us, then we’ll let you go free and your friend will go to prison for 7 years.

So you can be a bit naughty and snitch on your mate but, of course, you don’t know what your friends going to do or what you’re going to do. You’ve both been given these two options –  you can either cooperate with the police which means you go free and your friend gets 7 years in prison. You can just say absolutely nothing and get one year in prison. And what will happen if you both cooperate? Then, of course, the police are like you’re both just making this up if you both blaming the other person so you’ll each get 3 years if that happens.

So each of you has to decide whether or not you will stay silent and accept your 1 year. Or whether you will cooperate with the police and snitch on your friend. So I’d like you to both to write down whether you would cooperate with the police and snitch or stay completely silent. Of course, you’re not allowed to communicate here. You can’t discuss with each other what’s going on.

Georgia – I was going to say I want Peter to come back on the show so…

Tom – Hypothetical – bear that in mind.

Peter – Yeah. We’re sitting far enough apart we can’t see each other. We can see each other but not our bits of paper. I’ve written something down…

Georgia – I’ve written down something too…

Peter – Are we supposed to show it up like a game at Christmas or something?

Tom – I tell you what. We can do the maths here and work out what’s best. So if we do this first then we can see how you both thought about this problem.

Let’s start with Georgia and say let’s assume that Georgia has decided to cooperate with the police. So, from Peter’s point of view, Georgia has cooperated and snitched. So, at the moment, if Peter stays silent he is getting 7 years in jail because Georgia’s snitched. But if Peter cooperates with the police as well, his 7 years goes down to 3. So if Georgia snitched, Peter should also snitch. But if Georgia stayed silent, then that means at the moment Peter has 1 year but if he snitches he goes down from 1 to 0, whereas if he cooperates he goes up to 3.

Peter – And vice versa?

Tom – Yeah and vice versa. So if you think about it from an individual point of view you’re always reducing your jail time by snitching by cooperating with the police. But, if you think about it from a team perspective and you add up the total number of years in jail, the best option is for you both to stay silent because if you both stay silent you get 1 year each, a total of 2 years. Whereas if you both snitch, which is what you should do to maximise your own personal benefit here, your total time in jail is actually 6 years.

So this is basically how game theory works in a nutshell. It’s considering the individual benefit and individual decision as opposed to the team benefit and the team decision. So, let’s see what you actually have said…

Peter has decided to stay silent.

Georgia – Oh… I’m sorry, I snitched!

Tom – So Peter’s clearly far too nice…

Georgia – Please come back on the show. I’m sorry…

Peter – So the final result of that is how many years?

Tom – The final result would be Georgia would go free and unfortunately, Peter, you would be in jail for 7 years.

Peter – See – I’m a team player and Georgia…

Georgia – I just can’t go to prison.

Tom – But as you said, this is game theory. It’s the team dynamic versus the individual benefit.

Georgia – How do we use this? Who was right essentially? I’ve betrayed Peter massively but how do we use this?  If I’m ever in this situation should I be using game theory in this way?

Tom – It’s not so much that you should be using it, it’s more a way of modeling a decision making process. So, if you want to think about it purely from an individual level, then you’re using it to maximise your own individual benefit. Because by always snitching you’re always going to benefit yourself on an individual level but then, on a team level of course, you’re not. It’s not so much that you can use it to inform your decision making, it’s perhaps just being aware of what’s going on and it’s a way of modeling how we make decisions as rational decision makers.

Georgia – I see. So it’s you look at what you’ve done and this is a way to understand why. I know in nature game theory is used all the time to explain why animals do certain things the way the do so I guess it’s a really useful tool in that way.

Tom – Exactly. It’s just the idea of thinking things on an individual level and on a team level, there are different options there.

Georgia – Thank you, Tom for showing us the delights of game theory then. And thank you Peter – I’m very sorry I’ve put you in prison for 7 years.

Would Aliens Understand Maths?

Interview with Professor Ian Stewart for the Naked Scientists – listen here.

Ian – It’s a more difficult question than most people tend to think because we’ve been brought up with this idea as maths as the kind of universal truth. It’s somehow truer than anything else because it’s all perfectly logical and it follows from basic principles and so forth. Two plus two has to be equal to four, once you’ve decided what two, four, plus, and equals mean and there’s no real way round that. And all of that’s very true so I doubt you’d find aliens who don’t think that two plus two equals four, but you might find aliens who don’t really understand two, or plus, or equals.

In fact, Henri Poincare was one of the great French mathematicians at the end of the 19th century-beginning of the 20th century said “in the real world equals does not behave the way we think it does.” In mathematics, if A equals B, and B equals C, then A equals C. In the real world if we think A equals C actually they might be very slightly different but we can’t measure the difference. I suppose you could only measure things to within 1cm. If two things are ¾ cm apart, you think they’re in the same place. And if the second one is ¾ cm away from one of those, you think that’s in the same place as the other one. But then two of them could be 1½ cm apart from each other and you could tell they’re different. So, in the real world if A equals B and B equals C, then maybe A is different from C, and this is what Poincare said.

So mathematics is an idealisation of what’s out there and I think that different creatures from different environments with different ways of living, different concerns as to what’s important might set things up in a different way.

Tom – It almost sounds to me like you’re saying the maths we have on Earth has evolved from our environment on Earth and so the maths that aliens use would have evolved from their environment?

Ian – If you actually start thinking about it, what we consider to be absolutely fundamental mathematical concepts are very closely related to the kind of creatures we are, the kind of world we live in, and how we perceive that world. Our counting came from keeping track of discrete objects or events. If I’ve got a lot of sheep, I’m a farmer with a lot of sheep – say 15 sheep, it’s useful to know that I’ve got 15 sheep. I’ve got to be able to count up to 15. If one of them’s missing I will spot it. If you just say well you know I’ve got some sheep, it’s not too many, it’s not too few, and you lose one of your sheep you don’t realise it’s gone. A bit later you lose another sheep and still don’t realise it’s gone and suddenly you’re down to 2 or 3 sheep and you think something happened there. But most of it happened so long ago you can’t do anything about it.

So we count things. And counting the number of days in a month in the cycle of phases of the moon; that was very important to a lot of early civilisations. So the idea of going 1, 2, 3, 4 and so on is very basic to us, but creatures that lived in a completely different environment where there aren’t any discrete objects. I’m thinking of creatures that maybe float around in the atmosphere of a gas giant planet. They’re more like giant amoebas or something – they flow, everything flows. So you try to explain Pythagoras’s theorem to one of these creatures and say – take a triangle with corners at A, B, C, and it puts down A. Then you say now make another point B and it says OK I’ve got point B but A’s disappeared – it’s blown away on the gas on the wind.

There aren’t any rigid triangles. They might have real trouble with concepts like triangle. On the other hand, turbulent vortex flow is something they would think is absolutely trivial and straightforward. A child of whatever would understand, and we have really, really serious mathematical problems with it.

Tom – It sounds to me like maths is still maybe our best bet with communicating with aliens? I mean better than language or our culture surely in terms of what they might have a chance of understanding?

Ian – I think that’s probably right. I think to be a bit less imaginative about the possibilities and just say what’s actually likely. Maths has a kind of universality that communicating with language would be difficult. They don’t speak the language, they might not even hear sounds. Poetry probably isn’t a great idea; the works of Shakespeare would not particularly impress them. Music possibly, but think of listening to music from other human cultures. It’s very hard to get used to and no generation of parents understands the music that their kids like – teenagers are aliens. It’s fairly notorious musical tastes are very different.

If there are points of contact, it’s somewhere you can start from. Very likely aliens would not use base ten numbers. The might use some sort of number base because it’s quite a neat idea for encoding numbers. But we’ve got ten fingers, if you count thumbs and that’s probably where base ten comes from – there’s nothing special about ten. So an alien with seven tentacles might might work in base seven. But they would rapidly think oh, these guys are working in base one three, that’s one seven plus three – we call it ten, they call it thirteen. And we would rapidly say oh, they’re working in base seven and you can translate from on to the other very straightforwardly.

So differences in notation, differences in number base, differences in… We think triangles are basic; no they prefer rectangles, whatever. That wouldn’t be a barrier to communication, it would be something to puzzle out. If we’re on that level, there would be this great possibility that the aliens know some maths that we don’t,and we know some that they don’t. I’ll trade you the proof of Fermat’s last theorem if you can tell me how to prove the Riemann hypothesis. You could actually have interstellar trade in mathematical theorems.


Where does river water go?

It might seem like a simple question, but just think about it for a second… Water falls from the sky as rain, it flows over and under the ground and enters into a river. The river flows downstream, maybe passing through a few lakes along the way, until it reaches the ocean. Now what happens? The water has to enter the sea and will eventually be evaporated by heating from the sun and end up back in the atmosphere to form rain again. A lovely full-circle route called the water cycle – you probably learnt about it in Geography class. But, what if we could track a raindrop from the sky, into a river and then into the ocean. What would happen? Does it just flow into the ocean and then get mixed up and thrown around in the wind and waves? Or do tides drag it out further to sea? And what about the fact that the Earth is rotating? Perhaps not so simple after all…


Image credit:

This is essentially what my thesis is about. When water from a river enters the ocean, where does it go? It took me almost 4 years to figure it out – or more accurately to understand more about what’s actually happening… I certainly do not have all of the answers (as you’ll see). My plan is this series of articles is to try and explain 4 years’ worth of laboratory experiments, fieldwork, computer simulations and of course maths, so that anyone can understand what I’ve done and why it is important.

That’s probably as good a place to start as any: why is it important to understand more about where river water goes? Also known as the classic question faced by all researchers: why should we care? The grand big-picture answer is of course that by understanding more about the world around us, then and only then, can we begin to answer the fundamental questions about the meaning of life, the universe and our very being. That all sounds a little too Brain Cox to me so let’s try a simpler reason… pollution.

From waste outflows leaving factories to fertilisers used by farmers, it all flows into our rivers. And we want to know where this pollution will end up so that we can try to stop it causing too much damage. If we know where river water goes, then we know where pollution goes – easy (or so we hope). Pollution from fertilisers is a particular problem because it’s difficult to stop. If a factory is pumping out pollution into a river we can tell them to stop and to dispose of the waste properly by some other means. If we tell farmers to stop using fertilisers then they produce less crops, which means less food for us – hopefully you can see the issue.

The fertilisers used on crops seep into the soil and enter the underground water supply. This then flows into rivers, which flow into the sea. Fertilisers contain lots of nitrogen and this is great for growing plants – they love the stuff. The problem with having lots of nitrogen in the ocean is that it causes huge plankton blooms. Plankton are little plant-like things floating everywhere in the ocean, basically tiny sea plants. If you get lots of plankton blooming at the surface of the ocean, it blocks the sunlight from reaching the plankton beneath the surface and so they can’t photosynthesise to produce food, which means that they die. Lots of dead material means lots of bacteria. These little critters break down the dead plant material and when doing so use up oxygen in the water until eventually there’s none left. This is very bad news for fish – they need oxygen to breathe – and so they end up dying too. It’s basically a big circle of death which we scientists call eutrophication.

The good news is that if we know where the river water ends up and therefore where the fertiliser ends up, we can put measures in place to stop eutrophication from happening. So no more dead fish – hooray! At least not until they are caught in giant nets, but that’s a whole other kettle of fish… (pun very much intended).

So there you have it: knowing where river water goes means we can control pollution and stop fish from suffocating to death. And of course by understanding more about the world around us we can begin to answer the fundamental questions… nope I can’t do it. I’ll leave the star-gazing to Brain Cox.



Funbers 3, pi and 4


Three’s a crowd… or is it? Imagine you’re writing a story, how many lead characters would you have? One means a hero, two means a love interest, and three comes next. This is why three is so popular throughout history – it’s the first number that allows for a team, without any romance. It’s also everywhere in religion: Christianity has the Holy Trinity, of the Father, the Son and the Holy Spirit; Islam has the three holy cities of Mecca, Medina and Jerusalem; Buddhism has the three Treasures of Buddha, Darhm and Sangha; Toaism has three deities called the Three Pure Ones; Brahma, Shiva and Vishnu are the three Hindu Gods and even going a little old school there are the three Norse Norns: Urd, Verdandi and Skuld who weave the tapestry of our fate with each thread representing the life of a single person… certainly something to ponder.

3.142… – PI

Everyone’s favourite mathematical food and quite possibly the most famous mathematical constant. Mathematical constants are numbers that aren’t part of the usual number line and aren’t fractions but pop up everywhere in maths. I’ve already mentioned root 2, the Golden Ratio and e in earlier articles to name but a few. Physics loves a good constant too – the gravitational constant G is a classic, and hands up if you’ve heard of h-bar? (If you haven’t don’t worry physics isn’t as cool as maths). Going back to pi, it’s of course most famous for circles. Take any circle, measure its radius (the distance from the centre to the edge) and then the area of the circle is given by pr2 and the distance around the edge of the circle (also known as the circumference) is 2pr. The fact that this works for any circle, ever, anywhere, that has ever previously existed or ever will exist, is what makes pi such a special number. The only explanation is that it’s part of the fabric of the universe. And it doesn’t stop with circles, pi pops up everywhere in physics too. Einstein’s equations for general relativity, check. Newton’s law of gravity, check. Maxwell’s equations of electromagnetism, check. I could go on, but I’d better get onto the next number… You get the point; pi is big news.

So big in fact, that you can watch a video I made for Pi day (March 14th or 3-14 if you’re American) right here.

4 – FOUR

Four is associated with symmetry, balance and stability. A table has four legs, lots of animals also have four legs (cows, sheep, pigs, lions, tigers, aardvarks, hippopotamuses…) and not forgetting even we have four limbs. There’s also the way we use four to divide everything up. There are four seasons in a year: winter, spring, summer and autumn, a compass has four points: North, East, South, West and there are four parts of the day: morning, afternoon, evening and night. The ancient Greeks went one step further and divided up everything in the world into one of the four elements: earth, fire, water and air. And their doctors believed that your body was filled with four liquids (which they called humors): blood, yellow bile, black bile and phlegm. If you were ill in ancient Greece, chances are your doctor would remove some of one of these liquids as an apparent cure… leeches anyone?

Then there’s the four horsemen of the apocalypse. I was a big fan of the Darksiders video game series growing up so I know all about these bad boys. They are mentioned in the Bible, though it’s difficult to know exactly what each one represents – sounds like most of the Bible to me… One is definitely death, that’s certain. The other popular choices are conquest, war (pretty similar no?) famine, pestilence and plague. Yes, that does make 6, but hey it’s the Bible, anything goes…

Oh, and I almost forgot, maths loves four too. There are four main operations: addition, subtraction, multiplication and division. It reminds me of the classic chat-up line: me plus you, subtract clothes, divide those legs and lets multiply… (crickets I know, but at least I tried).


Breaking 2

The past weekend has been a big one for running: not only did Nike try and break the 2-hour barrier for a marathon; I ran my first half-marathon after a four-year hiatus due to a dodgy knee. Whilst Nike may have failed, I smashed it. Dragged around Skopje (Macedonia in case you were wondering) by my brother George, I managed to finish 25th in a PB of 1:26:52 – also clocking in as the highest UK finisher. But enough about me let’s talk numbers…

By numbers I mean specifically the number 2. On Saturday morning three professional athletes, Eliud Kipchoge, Lelisa Desisa and Zersenay Tadese, set off into the darkness (literally – they started before dawn) as they attempted the impossible: a sub 2-hour marathon. For those of you that don’t run this might not mean much so let’s do some maths and break it down. A marathon is the grand old distance of 26.2 miles, also known as ridiculously far. The legend goes that a messenger Philippides ran the distance from the battlefield of Marathon to Athens to announce that the Persians had been defeated. After having exclaimed “we have won” he then collapsed and died from exhaustion, having not only run the entire distance, but also having fought in the battle himself. Legend or not (and most historians believe it to be inaccurate), I can certainly believe it. Humans are not designed to run such large distances in one go. Professional athletes we see on television make it look easy, but just think about it for a second… 26.2 miles. That’s 42.2 kilometres, 42200 metres, 105 laps of a running track – that’s insane.

But the distance isn’t the half of it. The current men’s world record marathon time is 2:02:57 held by Dennis Kimetto from Kenya and set in Berlin in 2014. That works out at 4:42 per mile… for 26.2 of them. Roger Bannister famously ran a sub 4-minute mile in 1954 (exactly 63 years to the day before Nike’s sub-2-hour marathon attempt), which is still today an incredible feat, but that is just for one mile by itself. The marathon world record pace may be a little slower, but here we are talking about 26.2 of them, in a row, with no breaks. As I said before, insane.

If you don’t know your imperial measurements (who does these days), the world record pace is 2:55 per kilometre or 70 seconds per 400m/1 lap of a running track. If I remember correctly, and I probably don’t, my best time at high school running the 400m was about 70 seconds. That’s me as a 16-year old sprinting flat out. To think that long-distance runners do that for 105 laps in a row is mind-blowing. We can continue and break it down even further into 100m segments. One marathon is equal to 422 lots of 100m, and to break the world record you would need to run each of them in a time of 17 seconds. It might not sound too hard, running a 17-second 100m race, you could probably go out and do it right now if you are physically active, but that would be one. Try doing 421 more at the same pace, in a row, with no breaks… again, insane. Hopefully you get the picture by now.


Over the past 100 years the marathon world record time has been steadily decreasing, albeit at a slower pace in recent times (as seen on the graph above). Nike’s idea was to use everything at their disposal to plan the perfect race and go sub 2 hours for the first time in history. At the level of elite athletes, taking 2:57 off a world-record is pretty much unheard of; and this attempt was not only a test of human endurance, but of science and technology too. The team behind the ‘breaking 2’ attempt consisted of world-experts in the fields of bio-mechanics, coaching, design, engineering, materials development, nutrition, sports psychology and physiology. They were meticulous in their detail, from the choice of course at Monza race track in Italy due to its favourable weather conditions, to individually designed attire for each athlete suited perfectly to their running style and technique, nothing was taken for granted. The runners were led by a group of 30-pacers that were interchanged at regular intervals to ensure energy levels were maintained. They even formed an arrow shape, with a lead man at the front, followed by a line of 2 and then 3 runners in front of the athletes, to minimise the effect of the wind.

Beginning before dawn, the three athletes set off on the journey of their lives. 17 laps of the famous circuit later, only one remained. Eliud Kipchoge completed the marathon in 2:00:25 – some two-and-a-half minutes inside the world record* but not as fast as he had hoped. It is of course unfair to label the attempt as a failure – the original BBC headline was changed from ‘Nike fail at sub-two hour marathon attempt’ to ‘Eliud Kipchoge goes close to sub-two hour marathon’’ – as how can something so incredible be seen as a failure? Kipchoge ran 26.2 miles at an average pace of 4:36 per mile – that’s only 15% slower than Roger Bannister’s 4-minute mile and he did 26.2 of them in a row!

It really was an incredible effort from one of the best athletes of our time. You could see the pain on his face as he attempted a sprint finish; this was pushing the human body to its very limits and beyond. As Kipchoge said himself after the event, the next question is how can the team take another 1 second per mile off the time? Do that and the seemingly impossible will be realised – INSANE. If that doesn’t give you goose-bumps the video below certainly will…

*To clarify the attempt does not count as a world record because of the use of pacers that were not competing in the race (along with several other infringements of the official rules).

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