How can you show geometrically that 3 < π < 4?

Approximating Pi was a favourite pastime of many ancient mathematicians, none more so than Archimedes. Using his polygon approximation method we can get whole number bounds of 3 and 4 for the universal constant, with only high-school level geometry.

This is the latest question in the I Love Mathematics series where I answer the questions sent in and voted for by YOU. To vote for the next question that you want answered next remember to ‘like’ my Facebook page here.

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Equations Stripped: Normal Distribution

Stripping back the most important equations in maths so that everyone can understand…

The Normal Distribution is one of the most important in the world of probability, modelling everything from height and weight to salaries and number of offspring. It is used by advertisers to better target their products and by pharmaceutical companies to test the success of new drugs. It seems to fit almost any set of data, which is what makes it SO incredibly important…

You can watch all of the Equations Stripped series here.

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BBC News – Maryam Mirzakhani’s Legacy

Live interview on BBC News about the legacy of Iranian Mathematician Maryam Mirzakhani who tragically passed away today (July 15th 2017). She was the first female winner of the Fields Medal – the mathematical equivalent of the Nobel Prize.

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Crossing the desert

The fifth puzzle in the new feature from Tom Rocks Maths – check out the question below and send your answers to @tomrocksmaths on TwitterFacebook, Instagram or via the contact form on my website. The answer to the last puzzle can be found here.

You are responsible for driving an important person across the desert, but the cars that you have been given can only hold enough petrol to cover half of the distance. Being a desert, there are of course no petrol stations along the way. However, you have access to as many cars as you need and can transfer petrol between them.

What is the minimum number of cars that you will need and how can you complete the journey?

The answer will be posted in 2 weeks along with the next puzzle – good luck!

Play Nice!

Whoever said having fun is more important than winning was not a game theorist. Game theorists are mathematicians who study games, and how to win them. But they aren’t just interested in Snakes and Ladders – game theory also involves studying ‘games’ like nuclear standoffs, trade wars and even the competition of species as they evolve.

New research from the Institute of Science and Technology in Austria (http://dx.doi.org/10.1038/s41586-018-0277-x) might help us to use game theory for environmental good. Their findings look at perhaps the single most important problem we face in looking after the environment – ‘the tragedy of the commons’.

The tragedy of the commons plays out all around us, and relates to situations where everybody stands to benefit from damaging a useful shared resource. Everybody in the office exploits the ‘commons’ of the biscuit tin by taking a biscuit, but nobody can be bothered to go out and buy a new packet to keep the tin full. Eventually, the tin is empty, and everyone has to endure life without biscuits whilst someone looks for more. Such pointless suffering could have been avoided if only someone had acted sooner!

In a more serious setting, the tragedy of the commons can lead to catastrophic results. Take deforestation – the shrinking of the world’s forests as we use trees faster than they can grow back. It is in the individual interests of each logging company to spend all their time chopping down trees (which makes them money) and to waste none of it replanting them (which doesn’t). At least, in the short term. But over time, this clearly won’t work – the ‘commons’ of the world’s forests will be so damaged that everyone will lose out. Bad news for all you atmosphere fans out there.

The new research uses game theory to study the tragedy of the commons, to try and understand what we can do to prevent it. To stick with the logging example, the researchers treat logging companies as players competing in a series of very simple games, over and over, learning each other’s tactics. Each game is just a matter of choosing one of two options: Chop down trees without bothering to replant them, or take the time to replant them as well. Each time the choice is made, the company gets a reward depending on what they picked; they will get a bigger reward if they don’t use any of their time replanting. It looks like companies that are perfectly happy to drop-kick Dr Seuss’ orange defender-of-the-trees, the Lorax, are going to do better than their greener rivals.

At least, initially. The key to the new research is that in it, the games that have already been played affect the rewards up for grabs in the next game. If you keep choosing not to replant trees, then you may do better than your opponents in each game, but you’ll gradually make the rewards smaller and smaller as you start to run out of trees to cut down. So, you can’t just think about the profits to be won in today’s game – you have to think about what you’ll be playing for in tomorrow’s game too.

The researchers found that this makes a big difference to how companies will play. If previous games made no difference to the current game, then companies which don’t replant trees would do better than their replanting rivals. But, given that failing to replant the trees you cut down means worse prizes in the future, the companies which do replant end up doing a lot better than those that don’t bother. In other words, it pays to play nice.

The one catch to this is that the prizes have to get significantly worse when you choose not to replant. So, in practical terms, these findings suggest ways to make the ‘game’ of logging less environmentally devastating – by changing the rules. For instance, governments could pass laws which force any companies failing to replant trees to pay an increased tax on any future trees they cut down (or maybe pay for the Lorax’s extensive pension plan). This makes logging more like the game the researchers studied, where past choices quickly and significantly affect future rewards. So based on the researchers’ findings, such a law would make sure that doing the right thing and replanting trees is the better choice.

Yes, game theory is about winning. But by figuring out which rules reward the sort of people who go out and buy more biscuits for the tin, we can make sure the ‘winning tactic’ for the world’s most dangerous games is to play nice.

Joe Double

The World’s Smallest Knot

What do a clove hitch, a sheet bend and a sheepshank all have in common? They are of course, as any former scout will tell you, all knots. But I bet they couldn’t tie an 819 knot: at less than a millionth of a millimetre across, it’s the world’s smallest knot and has been tied by a team at the University of Manchester. They made the molecular tangle in a test tube using a sequence of carefully-controlled chemical reactions that used iron catalysts to bend and entwine short strings of carbon-rich molecules. I heard how from lead author David Leigh…

  • The knot is 192 atoms long with eight crossings and is the smallest, tightest knot ever tied.
  • The width of the knot is half a nanometre – less than one millionth of a millimetre or ten thousand times thinner than a human hair.
  • In mathematics, a knot describes a closed loop, which means that the knot here with its ends fused together, is still by definition a knot.
  • By viewing the positions of the atoms using X-ray crystallography, the knot can be seen to look like a four-leaf clover with extra strands wrapping around the outside of the leaves to generate the 8 crossings.
  • The knot is made using the technique of self-assembly where molecular strands are woven around metal ions, not too dissimilar to knitting.
  • The new technique used to make the knot could lead to a method of weaving molecular strands together to form stronger, lighter and more flexible materials.
  • In particular, Kevlar vest could be made much stronger by weaving the ‘rods’ of material together, rather than having them packed closely together like pencils in a pencil box, as is currently the case.

You can listen to the full interview for the Naked Scientists here.

Funbers 14

From plagues and lambs, to wives and rams the Bible loves a good reference to the number 14. And then of course there’s St Valentine and his quest to fill the world with love – not unlike a certain Gene Simmons from KISS…

You can listen to all of the Funbers episodes from BBC Radio Cambridgeshire and BBC Radio Oxford here.

England v Colombia Penalty Shootout

I was asked by the Daily Mirror to analyse the England football team’s penalty kicks against Colombia in the World Cup second round. You can find the key insights below and the full article online here.

unsaveable-zone

Image: Dr Ken Bray, University of Bath

Screen Shot 2018-07-09 at 13.32.28

Harry Kane – Kane’s very calm and confident in his walk up to the penalty spot showing that he has prepared well mentally. He carefully places the ball and adjusts his socks before firing low and hard into the bottom left-hand corner of the net. The keeper goes the right way but it’s too accurate and right in the corner of the ‘unsaveable zone’.

Marcus Rashford – A different approach on the walk up as he keeps his head down to make sure he doesn’t give anything away to the Colombia keeper. He curves his run-up to add extra disguise to the shot and puts it in almost exactly the same place as Harry Kane. Again, the Colombia keeper goes the right way but it’s too fast, too accurate and right in the bottom corner of the ‘unsaveable zone’.

Jordan Henderson – The ‘kick-ups’ on the walk to the penalty area show he’s nervous and the look on his face also hints at a lack of confidence. The placement of the shot is actually very good as he hits the ‘unsaveable zone’ to the left of the keeper, but his shot is a little higher than the previous two making it a more comfortable height for the goalie, and his wide run-up gives the game away as he opens his body to go to the right. If you look closely you’ll see that Ospina moves before Henderson kicks the ball which is why he’s able to reach beyond the ‘diving envelope’ and make the save.

Kieran Trippier – He has his head down and a look of complete focus on his face as he approaches the penalty spot. After a little glance up to make sure he knows where he’s going, he buries it in the top left corner in the perfect spot. Comparing Trippier’s penalty to the fourth Colombian taker, Uribe, who missed, it’s the use of the inside of his foot that makes all of the difference. Despite them both aiming for the top corner of the ‘unsaveable zone’, Uribe leant back and went with his laces making it less controlled than Trippier’s side foot. It’s also interesting that England’s nominated set piece taker went fourth in the line-up. No doubt, because Gareth Southgate knew that the fourth penalty would be key to victory as one that goalkeepers are likely to save.

Eric Dier – Positionally, probably the worst of the five England penalties as it was the closest to the centre of the goal and the edge of the ‘diving envelope’ which is within reach of Ospina. The key aspect of Dier’s penalty that allowed him to score was the fact that it was along the ground. Ospina dives the correct way, but can’t reach close enough to his body to make the save. Compare this to Jordan Henderson’s penalty, which was much closer to the corner, but at a more comfortable height for the save.

Summary:

  • 4 of the 5 penalties went to the left of the goalkeeper and were all scored, whereas the one that went to the right of the keeper was saved.
  • All of England’s penalty takers were right-footed.
  • 2 of the 5 penalty takers were substitutes, likely brought on to take a penalty in the shootout.
  • All of England’s penalties hit the ‘unsaveable zone’, maximising the chances of scoring. For Colombia only 2 of the 5 penalties hit the ‘unsaveable zone’.
  • Jordan Pickford saved the fifth and final penalty, demonstrating how it is more likely for a goalkeeper to make a save later in the shootout.

England benefitted from good preparation from the manager in selecting his line-up months in advance, aiming consistently for the ‘unsaveable zone’ which is the most difficult area for the goalkeeper to reach, and by preparing well mentally and taking their time with each shot. Ultimately, these 3 things were key to the victory.

New Scientist Live – Maths v Sport

Come and see my talk at New Scientist Live on September 23rd as I explain how to take the perfect penalty kick, investigate whether or not humans will ever break the 2-hour marathon barrier, and examine how the Earth’s rotation affects sports as varied as Golf, Rowing and Cricket.

Use code CRAWFORD10 for 10% off your ticket price – book here!

Music Taste Linked to Brain Type

How does the way you think influence the music you choose to listen to? Scientists at Cambridge University have developed a test that marries up a person’s personality traits including how empathic they are, and how systematically they think, with the tunes most likely to resonate with them. I went to see the lead researcher David Greenberg to discover what the test revealed about my own musical tastes…

David – The measure of empathy is called the empathy quotient and it’s a sixty-item measure that asks about how you interact in your daily life and your care for others, how you perceive emotion and react to emotion and thoughts of others. Another dimension is called systemising and systemising is the drive to construct, analyse and look at the rules that govern different aspects of the world.

Tom on the empathy quotient you scored a 56, and the average male scores around 30 so you were slightly above average on empathising. On systemising you scored very high – so you’re score is a 95 and the average male usually scores a 68.

Tom – Okay so that makes sense I guess – I do maths, I do see patterns in things and so this is sort of reflecting how I would have thought my brain worked.

David – It’s not too surprising because previous research has shown that males tend to score higher than females on systemising. And mathematicians score higher on systemising than for example students who are studying humanities.

Tom – And then once you’ve worked out how someone thinks, how did you then try to find out their musical preferences – do you say to them perhaps ‘here’s a list of band names who’d you like’?

David – No, so that’s been done previously where participants would just list how much they like a genre, but the problem with genres is that they’re so vast. If you take the rock genre in general, you have heavy metal, punk and you have bands like Metallica. But also in the rock genre you have Jeff Buckley or Jodie Mitchell and so there’s a vast difference. So we thought a more accurate way of doing it could be to just administer pieces of music to the participants: have them listen and then to indicate how much they liked each piece of music.

Tom – And so what did you find then? Once these participants have done this questionnaire you’ve worked out how they think – how did this affect their music choices?

David – What we found quite consistently over several studies was that empathisers in terms of the style of music that they liked, they were preferring music that was mellow and was from R&B, adult contemporary and soft rock genres. Whereas, systemisers were preferring music that was more intense and that was from the punk and heavy metal genres.

Tom – So what do I like? What did you find out about me?

David – You scored for example with mellow music or unpretentious styles which is from the folk genres or music that’s from classical or jazz, you scored average on those preference dimensions. But you scored the highest on intense music – so musical extracts that were from the punk, heavy metal and hard rock genres those were your favourites by far.

[MUSIC]

Tom – That was my favourite one that I listened to yesterday! I feel like I’ve been the perfect test student here! We’ve just been discussing exactly what type of music a systemiser should like and we’re just looking at my results here and I’ve nailed it to be honest!

And are there any applications for this beyond just figuring out which music people should and shouldn’t like?

David – A lot of research and there’s volumes of it has shown that music can be effective in music therapies. So, for example in terms of social skills or emotion recognition, we could use these results as a way of say teaching emotion recognition to children through music.

Tom – Based on my test results, play me the song that I should absolutely hate – I should leave the room I should dislike it that much!

[MUSIC]

…Yeah not liking that! That’s just so depressing I’m just not buying it.

David – But that’s the great thing about this study: there’s really no right or wrong answer. It’s just that people like different things and you can actually say that music is a mirror of the self in a way, it’s a reflective of who we are. And that our musical choices are a link or an expression of our mind, our personalities and the way we interact with the world.

You can listen to the full interview with the Naked Scientists here.

Funbers 13

The unluckiest number in the world… or is it? The Far East might have something to say on the matter… Plus, the Last Supper, the gallows and puberty.

You can listen to all of the Funbers episodes from BBC Radio Cambridgeshire and BBC Radio Oxford here.

Looking inside your heart

One of the tools available to doctors to see the heart in action is the echocardiogram. This uses ultrasound waves to image the heart as it beats, so the cardiologist can tell whether it’s contracting correctly and that the heart muscle is a healthy shape. I volunteered to be a guinea pig…

Clare – I’m Clare Ward-Jones; I work for Phillips Healthcare and my role is a cardiac ultrasound applications specialist. We’ve got a machine the ETHIC and that is the supreme ultrasound machine for cardiology. We can do a 2D scan on you, we can also do 3D images so we can get a 3D model of the heart.

  • The machine is the same as that used for a scan on a pregnant woman and relies on ultrasound beams which are sent out from the end of the probe and bounce back when they hit structures in the body.
  • Patients must lie on their left-hand side to bring their heart closer to the front of the chest meaning the ultrasound has to travel a shorter distance and therefore produces a clearer image on screen.
  • Electrodes are also applied to the patient to monitor the heart rate during the procedure – mine measured 87 which lies between the normal rate of 60-100 beats per minute.
  • Jelly is applied to the skin to remove any air between the probe and the skin which ultrasound cannot travel through.

Rick – My name is Rick Steeds; I’m a consultant cardiologist. I’m particularly interested in cardiovascular imaging and I’m the current President of the British Society of Echocardiography.

  • Echo sounds are very good at showing the structure of the heart; how strong the muscle is; whether the valves work; whether they leak or whether they’re narrowed, and whether there’s damage to the heart, for example, from a heart attack.
  • The images of my heart show two of the four chambers and the valves between them opening and closing.

You can listen to the full interview for the Naked Scientists here.

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