I was recently interviewed by Lucia Taboada for La Redada Podcast about my love of maths and how it is used in today’s world to model everything from penalty kicks to the next TV series you watch on Netflix. The interview was translated into Spanish for the actual podcast so I’ve also included the original recording of my answers in English – enjoy!
On your YouTube channel, you present science in an entertaining way. Why is maths so unpopular sometimes, maybe students are afraid of maths?
How would you define the importance of mathematics in our life?
Tom, I’m a huge supporter of a Spanish team called Celta de Vigo. You explain the possibilities using maths to improve the performance of football players. How can Celta de Vigo use this to improve? (unfortunately, we are now in the last positions)
Penalty kicks are a science? Can you predict them?
Have you been hired by any football team?
Do you think football teams should hire math workers?
You are a tutor in St John’s College at the University of Oxford where you teach maths to the first and second year undergraduates. Oxford is a traditional university – how are your methods received there?
You have some maths tattoos on your body, thats right? Explain them to us?
During my recent trip to Madrid to speak at the Residencia de Estudiantes, I was interviewed by national newspaper ‘El Mundo’ about my talk on the ‘Maths of Sport’ and my mission to popularise maths. The original interview can be found here.
Known as Tom Rocks Maths, the Oxford University scientist transforms boring formulas into fascinating models that he applies to sport to improve records and reduce errors.
MAR DE MIGUEL | Madrid
Football World Cup, 2018. 1-1 on the scoreboard. Spain plays its pass to the quarterfinals in the penalty round against Russia. Koke has failed one. Cheryshev is ready to throw. He scores. It’s up to Aspas. Expectation. Whistle. Launch and … Akinfeev stops it. We miss the game.
Could it have been avoided? The answer is Tom Crawford, l’ enfant terrible of numbers, a punkrocker in the court of mathematicians at the University of Oxford. And he explains it with a worn shirt, leather jacket, curled hair, piercing and tattoos. Because Crawford is not a common scientist. He is Tom Rocks Maths, an alternative researcher and communicator who transforms boring formulas into fascinating models that he applies to sports science, his second passion as a marathoner and a follower of Manchester United.
But, since all science is not exact, nor is Crawford a fortune teller, his predictions are based on data, taking into account all possible variables and, above all, on the highest probability of hitting. It is about getting ahead of the facts, of having all the necessary information to reduce errors and improve the records.
The mathematics of sport consist of “building models using data from the past to predict the future. When you don’t have them, you have to go to the field, contact the athletes and gather new information, ”explains Crawford in an interview with EL MUNDO after the talk he gave Tuesday in Madrid during the cycle of conferences ‘Mathematics in the Residence’, organized by ICMAT, the Student Residence and the Deputy Vice Presidency of Scientific Culture of the CSIC.
Madrid, 12 de noviembre de 2019. El matemático Tom Crawford posa en la Residencia de Estudiantes antes de una conferencia. Foto: Antonio Heredia
Win or lose penalties
The countries that best know how to throw penalties are Uruguay, Germany, Argentina and Brazil. Spain is not good, not bad. We are 50% among this list of experts and 50% of the worst, Mexico. We share media with France and Ireland. But how a team is better than another is not a matter of tradition or genetics, but numbers.
The first thing is to know how the players have responded before to the penalties, their statistics of failures and successes. According to Crawford, in the case of the 2018 World Cup, while Iniesta had four hits of five shots, Koke had zero of one and Aspas 16 of 17. The great surprise could have been given by Thiago, a substitute with a full in hits, four of four.
There is also a way to measure their stress responses with glasses that observe the movements of the eye. Footballers who are not immuted by pressure keep their eyes fixed and the most distracted move them. Knowing this in advance could decide the choice of a coach to choose the most focused players on those decisive penalties of a World Cup. “Football clubs now have entire teams of mathematicians and scientists who analyze all this data,” says Crawford.
But math doesn’t end there. In a goal you can make as many measures as your imagination, as Archimedes, to find the radius that indicates the exact area where you should place the ball without being stopped by the goalkeeper. It is called an insurmountable area and it depends, among other things, on the distance the goalkeeper moves in the shortest possible time from his position in the center. It looks like this: r2-2r(a+b+R)+a2+b2-R2. “These calculations are going to help you but they don’t guarantee that the penalty is perfect. In fact, the goalkeeper may also have trained against these formulas, ” Crawford alerts.
Roberto Carlos, the king of the Magnus effect
If penalties are a science, free kicks are not far behind. Defining its trajectory is one of Crawford’s favorite equations when the ball is given effect, as we have allways called it, which also has its scientific name: Magnus effect. “The ball that spins does not go in a straight line, because the rotation moves it to the side,” he said.
In this modality, for Crawford there is a master: Roberto Carlos, king of the Magnus effect in a match against France in 1997. It happened like this: he carefully placed the ball with his hands on the ground. He kicked. The ball passed over the barrier of players, turned in the air to the right, then to the left, hit a stick and entered as a stroke.
“I saw it when i was eight years old and I thought that it was impossible, that it was magic. But years later, using this equation to model Roberto Carlos’s shot, by entering the correct data (the speed of the ball, the distance to the door and the spin that applies to the ball) the formula accurately predicted that movement. It is still amazing. Although now I have an explanation that tells me that it did not break the barriers of physics.”
Marathon in less than two hours
In Tom Crawford’s mind there are not only favorite sport formulas but also graphics. And if there is one that drives him crazy it is the one that calculates, with a curve, when you will be able to run, with conventional methods, a marathon in less than 2 hours, something that it could happen between 2027 and 2035.
The record is owned by Kenyan Eliud Kipchoge, 2:01:39. He obtained it in Berlin in September 2018. In October of this year, the same athlete beat it at 1:59:40, but his feat was not accepted by the International Athletics Federation. “It has not been taken into account because they have broken the rules. As a new official mark, this record below two hours does not count, ”says Crawford.
How they did it? “Creating the perfect race,” says Crawford. And something else: a flat route in a straight line to go through the center of the track; a pair of shoes with carbon fiber that balances and saves 4% of energy; a tape on the leg with lumps (like golf balls) that create streams; a squad of escorts in V to cut the wind (called hares); a car that laser marks the ground so that these satellite corridors maintain the perfect position; a scanner that controls the muscle accumulation of carbohydrates and an enriched diet.
“Where you draw the line between what is due to the human element or an incredible shoe. What is the next? Putting rockets in our soles? ”Crawford wonders. We saw it in swimming a few years ago with high-tech swimsuits that reduce friction with water. They were even questioned for increasing buoyancy. With them, in some competitions 130 records were broken in just two years.
It is clear that mathematics helps to overcome tests and marks. However, in sports there are uncontrollable factors, such as the mental control of athletes, to disarm algorithms. “You can never add that factor to your models. You can never really predict a sport with total certainty. There are many unknown variables, ”reflects the English mathematician.
And, returning to the football game that we lost in 2018, would we have win if we knew the data in depth and having other players thrown the penalties that took us out of the World Cup? According to Crawford, we could have reduced the risk of losing, but this is something we will never know. What is highly certain is that their talks not only reinforce the devotion to sports, but also awake the mathematical vocation of the youngest students.
Following Manchester City’s penalty shootout victory over Liverpool in the Community Shield, I was asked by BBC Oxford to explain how scientists are trying to find the formula for the perfect penalty…
In October 2017, Dr Tom Crawford joined St Hugh’s as a Lecturer in Mathematics. He has since launched his own award-winning outreach programme via his website tomrocksmaths.com and in the process became a household name across Oxford University as the ‘Naked Mathematician’. Here, Tom looks back on the past year…
I arrived at St Hugh’s not really knowing what I was getting into to be completely honest. I’d left a stable and very enjoyable job as a science journalist working with the BBC, to take a leap into the unknown and go it alone in the world of maths communication and outreach. The plan was for the Lectureship at St Hugh’s to provide a monthly salary, whilst I attempted to do my best to make everyone love maths as much as I do. A fool’s errand perhaps to some, but one that I now realise I was born to do.
The ‘Naked Mathematician’ idea came out of my time with the Naked Scientists – a production company that specialises in broadcasting science news internationally via the radio and podcasts. The idea of the name was that we were stripping back science to the basics to make it easier to understand – much like Jamie Oliver and his ‘Naked Chef’ persona. Being predominantly a radio programme, it was relatively easy to leave the rest up to the listener’s imagination, but as I transitioned into video I realised that I could no longer hide behind suggestion and implication. If I was going to stick with the ‘Naked’ idea, it would have to be for real.
Fortunately, the more I thought about it, the more it made sense. Here I was, trying to take on the stereotype of maths as a boring, dreary, serious subject and I thought to myself ‘what’s the best way to make something less serious? Do it in your underwear of course!’ And so, the Naked Mathematician was born.
At the time of writing, the ‘Equations Stripped’ series has received over 100,000 views – that’s 100,000 people who have listened to some maths that they perhaps otherwise wouldn’t have, if it was presented in the usual lecture style. For me that’s a huge victory.
Of course, not all of my outreach work involves taking my clothes off – I’m not sure I’d be allowed in any schools for one! I also answer questions sent in by the viewers at home. The idea behind this is very simple: people send their questions in to me @tomrocksmaths and I select my favourite three which are then put to a vote on social media. The question with the most votes is the one that I answer in my next video. So far, we’ve had everything from ‘how many ping-pong balls would it take to raise the Titanic from the ocean floor?’ and ‘what is the best way to win at Monopoly?’ to much more mathematical themed questions such as ‘what is the Gamma Function?’ and ‘what are the most basic mathematical axioms?’ (I’ve included a few of the other votes below for you to have a guess at which question you think might have won – answers at the bottom.)
The key idea behind this project is that by allowing the audience to become a part of the process, they will hopefully feel more affinity to the subject, and ultimately take a greater interest in the video and the mathematical content that it contains. I’ve seen numerous examples of students sharing the vote with their friends to try to ensure that their question wins; or sharing the final video proud that they were the one who submitted the winning question. By generating passion, excitement and enthusiasm for the subject of maths, I hope to be able to improve its image in society, and I believe that small victories, such as a student sharing a maths-based post on social media, provide the first steps along the path towards achieving this goal.
Speaking of goals, I have to talk about ‘Maths v Sport’. It is by far the most popular of all of my talks, having featured this past year at the Cambridge Science Festival, the Oxford Maths Festival and the upcoming New Scientist Live event in September. It even resulted in me landing a role as the Daily Mirror’s ‘penalty kick expert’ when I was asked to analyse the England football team’s penalty shootout victory over Colombia in the last 16 of the World Cup! Most of the success of a penalty kick comes down to placement of the shot, with an 80% of a goal when aiming for the ‘unsaveable zone’, compared to only a 50% chance of success when aiming elsewhere.
Image courtesy of Ken Bray
In Maths v Sport I talk about three of my favourite sports – football, running and rowing – and the maths that we can use to analyse them. Can we predict where a free-kick will go before it’s taken? What is the fastest a human being can ever hope to run a marathon? Where is the best place in the world to attempt to break a rowing world record? Maths has all of the answers and some of them might just surprise you…
Another talk that has proved to be very popular is on the topic of ‘Ancient Greek Mathematicians’, which in true Tom Rocks Maths style involves a toga costume. The toga became infamous during the FameLab competition earlier this year, with my victory in the Oxford heats featured in the Oxford Mail. The competition requires scientists to explain a topic in their subject to an audience in a pub, in only 3 minutes. My thinking was that if I tell a pub full of punters that I’m going to talk about maths they won’t want to listen, but if I show up in a toga and start telling stories of deceit and murder from Ancient Greece then maybe I’ll keep their attention! This became the basis of the Ancient Greek Mathematicians talk where I discuss my favourite shapes, tell the story of a mathematician thrown overboard from a ship for being too clever, and explain what caused Archimedes to get so excited that he ran naked through the streets.
This summer has seen the expansion of the Tom Rocks Maths team with the addition of two undergraduate students as part of a summer research project in maths communication and outreach. St John’s undergraduate Kai Laddiman has been discussing machine learning and the problem of P vs NP using his background in computer science, while St Hugh’s maths and philosophy student Joe Double has been talking all things aliens whilst also telling us to play nice! Joe’s article in particular has proven to be real hit and was published by both Oxford Sparks and Science Oxford – well worth a read if you want to know how game theory can be used to help to reduce the problem of deforestation.
Looking forward to next year, I’m very excited to announce that the Funbers series with the BBC will be continuing. Now on its 25th episode, each week I take a look at a different number in more detail than anyone ever really should, to tell you everything you didn’t realise you’ve secretly always wanted to know about it. Highlights so far include Feigenbaum’s Constant and the fastest route into chaos, my favourite number ‘e’ and its link to finance, and the competition for the unluckiest number in the world between 8, 13 and 17.
The past year really has been quite the adventure and I can happily say I’ve enjoyed every minute of it. Everyone at St Hugh’s has been so welcoming and supportive of everything that I’m trying to do to make maths mainstream. I haven’t even mentioned my students who have been really fantastic and always happy to promote my work, and perhaps more importantly to tell me when things aren’t quite working!
The year ended with a really big surprise (at least to me) when I was selected as a joint-winner in the Outreach and Widening Participation category at the OxTALENT awards for my work with Tom Rocks Maths, and I can honestly say that such recognition would not have been possible without the support I have received from the college. I arrived at St Hugh’s not really knowing what to expect, and I can now say that I’ve found myself a family.
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.
Image: Dr Ken Bray, University of Bath
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.
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!
The 2018 World Cup in Russia kicks off today and so I bring you a special double-edition of Throwback Thursday looking at the science behind the perfect penalty kick… Fingers crossed the England players listen/read my website and we don’t lose to Germany in a penalty shootout (though let’s be honest we probably will).
Live interview with BBC Radio Cambridgeshire looking at the ‘unsaveable zone’ and the best way to mentally prepare for a penalty.
And if that wasn’t enough, here’s a full description of the ‘Penalty Kick Equation’…
For all of the footballers out there who have missed penalties recently, I thought I would explain the idea of the science behind the perfect penalty a little further, and in particular the maths equation that describes the movement of the ball. On the radio of course I couldn’t really describe the equation, so here it is:
If you’re not a mathematician it might look a little scary, but it’s really not too bad. The term on the left-hand side, D, gives the movement of the ball in the direction perpendicular to the direction in which the ball is kicked. In other words, how much the ball curves either left or right. This is what we want to know when a player is lining up to take a penalty, because knowing how much the ball will curl will tell us where it will end up. To work this out we need to input the variables of the system – basically use the information that we have about the kick and input it into the equation to get the result. It’s like one of those ‘function machines’ that teachers used to talk about at school: I input 4 into the ‘machine’ and it gives me 8, then I put in 5 and I get 10, what will happen if I input 6? The equation above works on the same idea, except we input a few different things and the result tells us how much the ball will curl.
So, what are the inputs on the right-hand side? The symbol p just represents the number 3.141… and it appears in the equation because footballs are round. Anytime we are using circles or spheres in maths, you can bet that p will pop up in the equations – it’s sort of its job. The ball itself is represented by R which gives the ball’s radius, i.e. how big it is, and the ball’s mass is given by m. We might expect that for a smaller ball or a lighter ball the amount it will curl will be different, so it is good to see these things are represented in the equation – sort of a sanity test if you will. The air that the ball is moving through is also important and this is represented by r, which is the density of the air. It will be pretty constant unless it’s a particularly humid or dry day.
Now, what else do you think might have an effect on how much the ball will curl? Well, surely it will depend on how hard the ball is kicked… correct. The velocity of the ball is given by v. The distance the ball has moved in the direction it is kicked is given by x, which is important as the ball will curl more over a long distance than it will if kicked only 1 metre from the goal. For a penalty this distance will be fixed at 12 yards or about 11m. The final variable is w – the angular velocity of the ball. This represents how fast the ball is spinning and you can think of it as how much ‘whip’ has been put on the ball by the player. Cristiano Ronaldo loves to hit them straight so w will be small, but for Beckham – aka the king of curl- w will be much larger. He did of course smash that one straight down the middle versus Argentina in 2002 though…
So there you have it. The maths equation that tells you how much a football will curl based on how hard you hit it and how much ‘whip’ you give it. Footballers often get a bad reputation for perhaps not being the brightest bunch, but every time they step up to take a free kick or a penalty they are pretty much doing this calculation in their head. Maybe they’re not quite so bad after all…
A couple of weeks ago I talked about the science behind the perfect penalty kick on BBC Radio Cambridgeshire, and lo and behold a few days later the Championship Playoff final went to penalties. I may have jinxed it – sorry. The penalty shootout was actually of a pretty high standard with 7 out of 10 penalties being scored, two saves and only one missing the target. Clearly Reading’s Liam Moore did not listen to my interview…
For all of the footballers out there who have missed penalties recently, I thought I would explain the idea of the science behind the perfect penalty a little further, and in particular the maths equation that describes the movement of the ball. On the radio of course I couldn’t really describe the equation, so here it is:
If you’re not a mathematician it might look a little scary, but it’s really not too bad. The term on the left-hand side, D, gives the movement of the ball in the direction perpendicular to the direction in which the ball is kicked. In other words, how much the ball curves either left or right. This is what we want to know when a player is lining up to take a penalty, because knowing how much the ball will curl will tell us where it will end up. To work this out we need to input the variables of the system – basically use the information that we have about the kick and input it into the equation to get the result. It’s like one of those ‘function machines’ that teachers used to talk about at school: I input 4 into the ‘machine’ and it gives me 8, then I put in 5 and I get 10, what will happen if I input 6? The equation above works on the same idea, except we input a few different things and the result tells us how much the ball will curl.
So, what are the inputs on the right-hand side? The symbol p just represents the number 3.141… and it appears in the equation because footballs are round. Anytime we are using circles or spheres in maths, you can bet that p will pop up in the equations – it’s sort of its job. The ball itself is represented by R which gives the ball’s radius, i.e. how big it is, and the ball’s mass is given by m. We might expect that for a smaller ball or a lighter ball the amount it will curl will be different, so it is good to see these things are represented in the equation – sort of a sanity test if you will. The air that the ball is moving through is also important and this is represented by r, which is the density of the air. It will be pretty constant unless it’s a particularly humid or dry day.
Now, what else do you think might have an effect on how much the ball will curl? Well, surely it will depend on how hard the ball is kicked… correct. The velocity of the ball is given by v. The distance the ball has moved in the direction it is kicked is given by x, which is important as the ball will curl more over a long distance than it will if kicked only 1 metre from the goal. For a penalty this distance will be fixed at 12 yards or about 11m. The final variable is w – the angular velocity of the ball. This represents how fast the ball is spinning and you can think of it as how much ‘whip’ has been put on the ball by the player. Cristiano Ronaldo loves to hit them straight so w will be small, but for Beckham – aka the king of curl- w will be much larger. He did of course smash that one straight down the middle versus Argentina in 2002 though…
So there you have it. The maths equation that tells you how much a football will curl based on how hard you hit it and how much ‘whip’ you give it. Footballers often get a bad reputation for perhaps not being the brightest bunch, but every time they step up to take a free kick or a penalty they are pretty much doing this calculation in their head. Maybe they’re not quite so bad after all…
