Maths, but not as you know it…
Season 2 of Tom Rocks Maths on Oxide Radio continues with maths and rock aplenty as we delve into the quantum realm to uncover the third Millennium Problem worth a cool $1 million… Plus the usual puzzle, fun facts about numbers you didn’t realise you’ve secretly always wanted to know, and brilliant music from Blink 182, My Chemical Romance and Asking Alexandria. This is maths, but not as you know it.
Image credit: Cath Ennis
Arriving at St John’s in 2008 to begin my study of mathematics, I was certain that within 4 years I would be working in the city as an actuary or an investment banker. Whilst I loved my subject, I saw it as means to obtain a good degree that would set me up for a career in finance. I’m not sure I could have been more wrong…
My current journey began towards the end of my second year, where I found myself enjoying the course so much that I wanted to continue to do so for as long as possible. This led me to research PhD programmes in the UK and the US, and I was fortunate enough to be offered a place to study Applied Maths at the University of Cambridge in 2012. During my time at Oxford, I found myself straying further and further into the territory of applied maths, culminating in a fourth-year course in fluid mechanics – the study of how fluids such as water, air and ice move around. This ultimately led to my PhD topic at Cambridge: where does river water go when it enters the ocean? (If you’re interested to find out more I’ve written a series of articles here explaining my thesis in simple terms.)
As part of my PhD I conducted experiments, worked on equations and even took part in a research cruise to the Southern Ocean. It was on my return from 6 weeks at sea that I had my first taste of the media industry via a 2-month internship with the Naked Scientists. I would spend each day searching out the most interesting breaking science research, before arranging an interview with the author for BBC radio. It was great fun and I learnt so much in so many different fields that I was instantly hooked. Upon completion of my PhD I went to work with the Naked Scientists full time creating a series of maths videos looking at everything from beehives and surfing, to artwork and criminals. You can watch a short trailer for the Naked Maths series below.
My work with the BBC and the media in general ultimately led me to my current position as a Mathematics Tutor at three Oxford colleges: St John’s, St Hugh’s and St Edmund Hall. This may not sound like the media industry, but the flexibility of the position has allowed me to work on several projects, including launching my website and my YouTube channel @tomrocksmaths where I am currently running two ongoing series. In the first, Equations Stripped, I strip back the most important equations in maths layer-by-layer; and for the second series in partnership with the website I Love Mathematics, I answer the questions sent in and voted for by students and maths-enthusiasts across the world.
Alongside my online videos, I am also writing a book discussing the maths of Pokémon – Pokémaths – and have a weekly show with BBC radio called ‘Funbers’ where I tell you the fun facts about numbers that you didn’t realise you’ve secretly always wanted to know. I have also recently presented at conferences in the US and India and hold regular talks at schools and universities, including for the Oxford Invariants and the Maths in Action series at Warwick University where I faced my biggest audience yet of 1200.
Looking back at my time at St John’s, I never would have imagined a career in the media industry lay before me, but the skills, experience and relationships that I formed there have undoubtedly helped to guide me along this path. I think it just goes to show that Maths is possibly the most universal of all subjects and really can lead to a career in any industry.
You can follow Tom on Twitter, Facebook and Instagram @tomrocksmaths for the latest updates.
Oxplore – the University of Oxford’s digital outreach portal – has recently reached its 50th BIG question! To celebrate we’ll be hosting a special livestream debate at 2pm on March 29th which you can join for FREE by registering here.
If you’re not already excited (and trust me you really should be), then here are some of my favourite highlights from the live events so far to get you in the mood!
On Wednesday March 13th I’ll be presenting my research to MP’s at the Houses of Parliament in the final of the STEM for Britain Competition. You can find my research poster on modelling the spread of pollution in the oceans here.
Read coverage of my entry by the Oxford Maths Institute, St Edmund Hall, St Hugh’s College and the Warrington Guardian. The press release from the London Mathematical Society is also copied below.
Dr Tom Crawford, 29, a mathematician at Oxford University hailing from Warrington, is attending Parliament to present his mathematics research to a range of politicians and a panel of expert judges, as part of STEM for BRITAIN on Wednesday 13th March.
Tom’s poster on research about the spread of pollution in the ocean will be judged against dozens of other scientists’ research in the only national competition of its kind.
Tom was shortlisted from hundreds of applicants to appear in Parliament.
On presenting his research in Parliament, he said, “I want to bring maths to as wide an audience as possible and having the opportunity to talk about my work with MP’s – and hopefully show them that maths isn’t as scary as they might think – is fantastic!”
Stephen Metcalfe MP, Chairman of the Parliamentary and Scientific Committee, said: “This annual competition is an important date in the parliamentary calendar because it gives MPs an opportunity to speak to a wide range of the country’s best young researchers.
“These early career engineers, mathematicians and scientists are the architects of our future and STEM for BRITAIN is politicians’ best opportunity to meet them and understand their work.”
Tom’s research has been entered into the mathematical sciences session of the competition, which will end in a gold, silver and bronze prize-giving ceremony.
Judged by leading academics, the gold medalist receives £2,000, while silver and bronze receive £1,250 and £750 respectively.
The Parliamentary and Scientific Committee runs the event in collaboration with the Royal Academy of Engineering, the Royal Society of Chemistry, the Institute of Physics, the Royal Society of Biology, The Physiological Society and the Council for the Mathematical Sciences, with financial support from the Clay Mathematics Institute, United Kingdom Research and Innovation, Warwick Manufacturing Group, Society of Chemical Industry, the Nutrition Society, Institute of Biomedical Science the Heilbronn Institute for Mathematical Research, and the Comino Foundation.
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.
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.
You can find all of Tom’s outreach material on his website tomrocksmaths.com and you can follow all of his activities on social media via Twitter, Facebook, YouTube and Instagram.
Answers to votes (watch by clicking the links):
Cast your mind back to the summer of 2018… we saw the warmest ever weather in the UK, Brexit was not yet a complete and utter disaster, and seemingly against all the odds the England football team reached the semi-finals of the World Cup for the first time since 1990. No doubt the team had a huge celebration together afterwards – but it wouldn’t be the first time that two of them have celebrated an occasion at the same time. As well as playing together at the heart of England’s defence, Manchester City duo Kyle Walker and John Stones also share the same birthday! Stones was born on 28th May 1994, making him 24 years old; Walker was born on the same day in 1990, meaning that he is exactly four years older than his teammate. How strange! Or is it…?
On the face of it, it seems quite surprising that in an England squad of just 23 players, two of them happen to share a birthday. However, as we’re about to see, this isn’t a freakish coincidence – maths says that it’s quite likely! What we’re talking about here is commonly known as the birthday problem: if there are a group of people of a certain size, what is the likelihood that at least two of them have the same birthday?
Let’s start by saying that we have a group of N people, and assume that birthdays are equally likely on every day of the year. (There is some evidence to suggest that this isn’t the case for top athletes; some say that they tend to be born early in the school year, such as around September in England. This is because they are slightly older than the other children in the year, and so they have a slight head-start in their physical development. However, we don’t want to make things too complicated, so we’ll ignore that for now.)
The easiest way to think about the problem is to first try to work out what the probability is that none of the N people share a birthday. Suppose our N people walk into a room, that is empty at first, one at a time. When the first person walks in, it’s obvious that they don’t share a birthday with anyone else in the room, because there isn’t anyone else. Therefore, they have the maximum probability of not sharing a birthday with anyone else in the room, which is 1.
Now think of the second person who walks in. The only way that they could share a birthday with someone in the room is if it happens to be exactly the same day as the first person. That means there is a 1 in 365 chance that they do share a birthday, so there is a 364 in 365 chance that they don’t.
Suppose that the first two birthdays don’t match, and then the third person walks in. They now have 2 days that they can’t share a birthday with, so there are 363 possible choices out of 365. Because we assumed that the first two didn’t match, we multiply the probabilities, so now the chance that none of them share a birthday is (364/365) * (363/365).
We can repeat this process until we get to our final person, number N. For example, the fourth person has 3 birthdays that they cannot share, so we multiply by a chance of 362/365; the fifth person has 4 days to avoid, so we include a probability of 361/365… By the time the Nth person walks in, there are N-1 people already in the room, so there are N-1 days that their birthday cannot fall on. This leaves them with 365-(N-1) possibilities out of 365.
To work out the total probability, we multiply all of these terms together which gives the likelihood that none of the N people share a birthday as
1 * (364/365) * (363/365) * (362/365) * … * ((365-(N-1))/365).
You might be thinking that this still looks like quite a big probability that none of them share a birthday, because all of the terms are very close to 1. But, if we try some values of N in a calculator, then it tells a very different story. (The percentages are calculated by finding the probability from the equation above and multiplying by 100.)
When N = 10, we get an 88% chance that none of them share a birthday. However, this drops down to 59% when there are N = 20 people. When we get to N = 23, the number of players in the England squad, the probability reaches just under 50%. That means that, incredibly, the likelihood that at least two of the 23 people share a birthday is just bigger than 50%!
So, in a random group of 23 people, it’s more likely than not that two of them share a birthday! This seems very strange at first; surely you’d need more than 23 people for a shared birthday to be more likely than not?! This is why the problem is commonly known as the birthday paradox – it might be very hard to get your head around, but the maths doesn’t lie!
Perhaps, in order to convince ourselves, we should look at some real-life examples. This is where the World Cup squads come into play: each team is restricted to bringing 23 players to the tournament. (We’ve seen that number before…) If our calculations above are correct, then if we picked any one of the World Cup squads, there would be roughly a 50:50 chance that at least two of the squad members share a birthday, which means that out of all of the squads that went to Russia, we would expect about half of them to have a birthday match. Well, let’s take a look…
Of the 32 teams, which were divided into 8 groups of 4, the following teams have at least one pair of players who share a birthday:
| Group A | Russia |
| Group B | Iran, Morocco, Portugal, Spain |
| Group C | Australia, France, Peru |
| Group D | Croatia, Nigeria |
| Group E | Brazil, Costa Rica |
| Group F | Germany, South Korea |
| Group G | England |
| Group H | Poland |
So, not only is there at least one team in every group with a birthday match, but if we count the total, there are 16 squads with a shared birthday pair – exactly half of the teams! The experimental results have matched up with the mathematical theory to perfection. Hopefully that’s enough to convince you that our calculations were indeed sound!
A slightly different question that you might ask is as follows: if I am in a group with a certain number of people, what are the chances that at least one of them shares my birthday? Is it the same idea? What we have worked out above is the probability that any two people in the room share a birthday (or rather, we worked out the opposite, but we can find the right answer from our working). Note that the pair doesn’t necessarily include you; it’s a lot more likely that it’s some other pair in the group.
In order to work out the answer to this similar sounding question, we work the other way around again, by calculating the probability that none of the N people share my birthday. For each of the N people, there is only one birthday that they cannot have, and that is mine (14th November, in case you were wondering), which means there are 364 out of 365 possibilities for each person. We no longer care whether their birthdays match up; we only care if they match with mine. So each person has a 364/365 chance of not sharing my birthday; and the overall probability is just 364/365 * 364/365 * … * 364/365, N times, which we write as (364/365)N.
Once again, we can plug some values of N into a calculator: N = 10 gives a 97% chance that no-one else has my birthday. For N = 50 the probability is still very high: there is an 87% chance that none of these 50 people have the same birthday as me. N = 100 gives 76%; N = 200 gives 58%; you have to go all the way to N = 253 before the probability dips below 50%, and it becomes more likely than not that at least one person will celebrate their birthday with me.
Applying this idea to all 736 players (32 squads of 23 players) involved in the World Cup, we should expect around 3 of them to have been born on the same day as me – 14th November. And I am very happy to confirm that France’s Samuel Umtiti, Switzerland’s Roman Burki, and Belgium’s Thomas Vermaelen all have what is undoubtedly the best birthday of the year… Two similar problems with two very different solutions!
You can check which footballers share a birthday with you at www.famousbirthdays.com/date/monthDD-soccerplayer.html, where you enter the month in words and the day in numbers (no preceding zero required).
Kai Laddiman
2018 Abel Prize Laureate Robert Langlands explains his work in the context of theorems versus theories. The second in a series of videos documenting my experience at the 2018 Abel Prize week in Oslo.
With thanks to the Norwegian Academy of Science and Letters for kindly providing me with a scholarship.
The author H. P. Lovecraft often described his fictional alien worlds as having ‘Non-Euclidean Geometry’, but what exactly is this? And would it really break our brains?
Produced by Tom Rocks Maths intern Joe Double, with assistance from Tom Crawford. Thanks to the Oxford University Society East Kent Branch for funding the placement.
Esther Lafferty meets Dr Tom Crawford in the surprisingly large and leafy grounds of St Hugh’s College Oxford as the leaves begin to fall from the trees. It’s a far cry from the northern town of Warrington where he grew up.
Tom is a lecturer in maths at St Hugh’s, where, defying all ‘mathematics lecturer’ stereotypes with his football fanaticism, piercings, tattoos, and wannabe rock musician attitude, he makes maths understandable, relevant and fun.
‘It was always maths that kept me captivated,’ he explains, ‘ever since I was seven or eight. I remember clearly a moment in school where we’d been taught long multiplication and set a series of questions in the textbook: I did them all and then kept going right to the end of the book because I was enjoying it so much! It was a bit of a surprise to my teacher because I could be naughty in class during other subjects, messing around once I’d finished whatever task we’d been set, but I’ve loved numbers for as long as I can remember and I still find the same satisfaction in them now. There’s such a clarity with numbers – there’s a right or else it’s wrong. In English or History you can write an essay packed with opinion and interpretation and however fascinating it might be, there are lots of grey areas, whereas maths is very black and white. I like that.’
‘My parents both left school at sixteen for various reasons but they appreciated the value of education. My mum worked in a bank so she perhaps had an underlying interest in numbers but it wasn’t something I was aware of. I went to the local school and was lucky enough to be one of the clever children but it wasn’t until I got my GCSE results [10 A*s] that the idea of Oxford or Cambridge was suggested to me. I would never have thought to consider it otherwise.
‘I remember coming down for an interview in Oxford, at St John’s, arriving late on a Sunday night and the following morning I took a stroll around the college grounds – I could feel the history and traditions in the old buildings and it was awesome. I really wanted to be part of everything it represented. I thought it would be so cool to study here so I was very excited when I was offered a place to read maths.
‘Studying in Oxford I found I was most interested in applied maths, the maths that underpins physics and engineering for example. ‘Pure’ maths can be very abstract whereas I prefer to be able to visualise the problems I am trying to solve and then when you work out the answer, there’s a sudden feeling when you just know it’s right.’
In his second year, Tom became interested in outreach work, volunteering to take the excitement of maths into secondary schools under the tutelage of Prof Marcus Du Sautoy OBE as one of Marcus’s Marvellous Mathematicians (or M3), a group who work to increase the public understanding of science.
‘I went to China one summer to teach sixth formers and it was great to have the freedom to talk about so many different topics. I spent another summer in an actuary’s office because I was told that was the way to make real money out of maths – it was a starkly different experience. I realised I was not at all cut out for a suit and a screen!’ Tom smiles. ‘I am a real people-person and get a real buzz from showing everyone and anyone that you can enjoy maths, and that it is interesting and relevant. I love the subject so much and I think numbers get a bad press for being dull and difficult and yet they underpin pretty much everything in the whole universe. They can explain almost everything and you’ll find maths in topics from the weather to the dinosaurs.
Take something like the circus for example – hula-hoops spinning and circles in the ring, and then the trapeze is all about trigonometry: the lengths and angles of the triangle. Those sequinned trapeze artists are working out the distances and directions they need to leap as they traverse between trapezes and its maths that stops them plummeting to the floor!’
Having spent four years in Oxford Tom then spent five years at Cambridge University looking at the flow of river water when it enters the sea, researching the fluid dynamics of air, ice and water, and conducting fieldwork in the Antarctic confined to a boat for six weeks taking various measurements in sub-zero temperatures. You’d never expect a mathematician to be storm-chasing force 11 gales in a furry-hooded parka, but to get the data needed to help to improve our predictions of climate change, that was what had to be done!
Tom also spent a year as part of a production group known as the Naked Scientists, a team of scientists, doctors and communicators whose passion is to help the general public to understand and engage with the worlds of science, technology and medicine. The skills he obtained allowed him to kick-start his own maths communication programme Tom Rocks Maths, where he brings his own enthusiasm and inspiring ideas to a new generation alongside his lectureship in maths at St Hugh’s.
A keen footballer (and a massive Manchester United fan) it’s no surprise Tom has turned his thoughts to football and as part of IF Oxford, the science and ideas festival taking over Oxford city centre in October, Tom is presenting a free interactive talk (recommend for age twelve and over) on Maths versus Sport – covering how do you take the perfect penalty kick? What is the limit of human endurance – can we predict the fastest marathon time that will ever be achieved? And over a 2km race in a rowing eight, does the rotation of the earth really make a difference? Expect to be surprised by the answers.
Esther Lafferty, OX Magazine
The original article can be found here.
