Make your own Pi

Here’s a little something to celebrate Pi Day 2020 – originally written for the St Edmund Hall blog.

March 14^th is Pi Day, and as of 2020 is also the official UNESCO International Day of Mathematics. You may be wondering what’s so special about a seemingly random day in the middle of March, and if you’re not from the US then you’re completely right to do so. The key is in the date. March 14^th is written as 3-14 using the US system, which just so happens to be the first three digits of the number Pi which is 3.14 to two decimal places. If we use the UK system, then we’d need 31-4, or the 31^st April, which unfortunately isn’t a real date in the Gregorian calendar. So, March 14^th it is.

But why Pi? Even as a mathematician this might seem a random choice of number to represent the International Day of Mathematics, at least until you ask yourself the following question: which single well-known number best represents the field of mathematics? As an applied mathematician (not too dissimilar to a physicist or engineer) my choice would be the number e – Euler’s number. It’s certainly a great representation of all-things calculus (and therefore pretty much any equation in physics), but well-known outside of the mathematical community? I’m not so sure. And herein lies the reasoning behind the choice of Pi. There may be more important numbers (and please do let me know which one you’d pick if in charge), but better-known I highly doubt it. So, Pi it is.

Now we’ve got a better feel for the day named after the number, let’s talk about Pi itself. You may know it in terms of circles, but it has the rather fantastic knack of cropping up in the most unexpected places… Quantum Theory? check. Einstein’s Theory of Relativity? Check. Newton’s Law of Gravity? Check. Three of the most important theories we use to explain the universe, and each of them has a formula containing the number Pi.

Quantum: Heisenberg’s Uncertainty Principle

Relativity: Einstein Field Equations

Gravity: Newton’s Gravitational Constant

This ‘superhero-like’ ability to appear everywhere means that we can have a lot of fun with how we define the number Pi. The standard definition is to use circles: the perimeter of the circle or circumference, c, is equal to Pi multiplied by the distance across the circle passing through the centre or the diameter, d.

We can make things a little more complicated by thinking about the other circle-based formulas that contain Pi and using them to define its value instead. For example, the volume of a sphere, V, is equal to 4/3 times Pi times the radius, r, cubed. Rearranging, we define Pi as follows:

Now, here’s the best bit. Since Pi appears in the formula for the volume of a sphere, it means that any calculation involving the need to work out the volume of a spherical object will include the number Pi, and that means we can rearrange any such formula to get a new definition. Here’s one for you:

Don’t believe me? Here’s how it works…

We want to know the answer to the question: how many ping-pong balls will it take to lift the Titanic from the ocean floor? The idea being that each ping-pong ball floats, and therefore has a positive buoyancy force (don’t worry too much about exactly what that means other than the fact that objects which float have a positive buoyancy force and those which sink have a negative buoyancy force). If we calculate the buoyancy force on a single ping-pong ball, then that will in fact tell us the amount of weight that each individual ball is able to support before it sinks. Think of doing the following experiment with a boat. If the boat is empty, then it will float. As you start to add weight, it moves down lower and lower into the water until eventually you’ve added so much weight that it is completely submerged and sinks down to the bottom of the lake. The total amount of weight that you added up until the moment before it’s submerged is equal to the maximum amount of weight that the boat is able to support. That weight is the positive buoyancy force of the empty boat.

In our boat experiment we can keep track of the weight we are adding to the boat, but with a ping-pong ball it isn’t quite so simple. Instead, we have to calculate the buoyancy force using a neat idea called ‘Archimedes Principle’. We don’t need to worry about the exact details, just that we have a formula for the buoyancy force courtesy of Archimedes (yes, the guy that ran naked through the streets of Ancient Greece). Archimedes Principle tells us that the buoyancy force of an object is equal to the weight of water displaced by the same object. And what this means in practice, is we just need to multiply the volume of a ping-pong ball by the density of the Atlantic Ocean to get the weight that a single ping-pong ball can support:

Volume = 4/3 x Pi x radius³ = 4/3 x 3.14 x 2³ = 33.5 cm³

Density of Atlantic Ocean = 1.027 g/cm³

Weight supported by 1 ping-pong ball = 33.5 x 1.027 = 34.4 g

At this point we must remember to subtract the weight of the ping-pong ball itself, to give the amount that we can ‘add’ before it starts to sink:

Total weight that can be lifted by 1 ping-pong ball = 34.4 – 2.7 = 31.7 g

Finally, the weight of the Titanic is 47,500,000,000 g and so the total number of ping-pong balls required to lift it is given by:

47,500,000,000 / 31.7 = 1.498 billion

(I’m aware I’ve taken you on a slight detour from our original topic of Pi, but I hope you’ve at least been able to follow the main ideas of the calculation. If you would like to see more details of how it all works, then you can watch me run through the solution in full in the video below.)

So, with only 1.498 billion ping-pong balls we can lift the Titanic from the depths of the Atlantic Ocean and display it for all to see in a museum. If only it were so simple… Whilst the calculation is itself entirely correct, there’s something we’ve missed. The idea of using ping-pong balls was a real suggestion put forward by a group of scientists in the 1970’s, which needless to say did not happen. If you think you have an idea why get in touch via the contact form here.

This is no doubt a rather surprising and hopefully interesting result, but remember we only got to ping-pong balls and the Titanic because we were thinking about fun ways to define the number Pi. Looking back through our calculations, you’ll see that Pi appeared when working out the volume of a ping-pong ball, and so by putting all of our calculations together and rearranging the equation (like we did for the volume of a sphere) we end up with the brilliant – and my all-time favourite – definition (shown again because I love it so much):

With Pi Day and the first ever International Day of Mathematics fast approaching, why not celebrate the wonderful world of numbers by creating your very own definition of Pi. As you’ve seen with the example above, all you need is a calculation involving circles or spheres and some rearranging to get the tastiest number all by itself. Give it a go and let me know how you get on! @tomrocksmaths

March 16, 2020 0

Funbers 22, 23 and 24

Fun facts about numbers that you didn’t realise you’ve secretly always wanted to know…

22 – TWENTY-TWO

Coming in hot, 22 happens to be one of my favourite numbers – if you divide it by 7 you get about 3.142, which is a handy way of getting close to pi without having to remember all the digits! Then of course there’s Joseph Heller’s famous novel Catch-22. In the book, Catch-22 is the Air Force policy which says that bomber pilots can only stop flying planes if they are declared insane. But like the name suggests, there’s a catch. Catch-22 says that asking for a mental evaluation to get declared insane is proof that you aren’t in fact insane. So technically, there’s a way to get out of flying more bombing runs… but if you try it, you get sent right back out in the next plane!

Twenty-two also pops up in the kitchen. Normally, if you are slicing a pizza using 6 cuts, you’d do it neatly and end up with 12 even slices – much like the numbers on a clock face. But if you were a lazy pizza chef and just sliced randomly, you could end up cutting slices in half and ending up with more pieces. And it turns out, the most pieces you can end up with after 6 cuts is, you guessed it, 22!

On a darker note, 22 was also the lucky number of the Haitian voodoo dictator Francois “Papa Doc” Duvalier. Papa Doc started studying voodoo folklore to spread rumours that he had supernatural powers, which let him rule through fear. But eventually, he started believing the rumours himself. He would only go outside his palace on the 22nd of the month, because he thought he was guarded by voodoo spirits on that lucky day. He even claimed to have killed JFK, whose assassination was on the 22nd of November 1963, supposedly by stabbing a voodoo doll of him 2222 times that morning…

23 – TWENTY-THREE

For 23 we’re going back to maths, and specifically prime numbers. A prime number remember, is one that can only be divided by itself and one without giving any remainder. Twenty-three has the unique property of being the smallest prime number which is not a ‘twin prime’ – that is a prime number which does not have another one within two spaces of it on the number-line. For example, 3, 5 and 7 are all close friends, while 11 and 13 go together. 17 is next to 19, but the nearest prime number to 23 is either four places below at 19, or six places above at 29, making it the smallest prime number to not have the ‘twin’ property.

Twenty-three is also big for birthdays. Not because the age of 23 is particularly special (although being the age mentioned in my favourite song – Blink 182’s ‘what’s my age again?’ – I do have a soft spot for it), but because of its appearance in the ‘Birthday Paradox’. The complete explanation is a little too long for Funbers, but in short it says that if you choose 23 people at random and put them in a room together, there is a greater than 50% chance that 2 of them share the same birthday. If that sounds too crazy to believe, check out a full explanation here from one of my students who applied it to the 23-man England squad for the 2018 Football World Cup. Now to enjoy some classic pop punk: “Nobody likes you when you’re 23…”

24 – TWENTY-FOUR

Who remembers Avogadro’s constant for the number of atoms contained in one mole of a substance from high school Chemistry? No, me neither. But, a great way to approximate it is using 24 factorial – or 24! in mathematical notation. The factorial function (or exclamation mark) tells you to multiply all of the numbers less than 24 together. So, 24! is equal to 24 x 23 x 22 x 21 x 20 x 19 x … x 2 x 1, also known as an incredibly large number. It’s about 3% larger than Avogadro’s constant, but certainly easier than remembering 6.02214076 x 10²³.

Twenty-four also represents the number of carats in pure gold, the number of letters in the Greek alphabet (ancient and modern) and the number of points on a backgammon board. Mathematically, 24 is the smallest number with exactly 8 numbers that divide it – can you name them? And, it’s equal to exactly 4 factorial: 4! = 4 x 3 x 2 x 1 = 24. Last but not least, where would we be without the 24 hour day – or to be precise 24 hours plus or minus a few milliseconds to be completely exact…

	Day length
Yesterday	24 hours -0.46 ms
Today	24 hours -0.39 ms
Tomorrow	24 hours -0.35 ms
Shortest 2019	24 hours -0.95 ms
Longest 2019	24 hours +1.67 ms
Last Year Average	24 hours +0.69 ms

August 6, 2019 0

Maths, but not as you know it… (St Hugh’s College Oxford Magazine)

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.

unsaveable-zone — 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 25^th 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):

What is the probability I have the same PIN as someone else?
How does modular arithmetic work?
What would be the Earth’s gravitational field if it were hollow?
What are grad, div and curl? COMING SOON

February 19, 2019 1

Ancient Greek Mathematicians

A new feature from Tom Rocks Maths – a weekly maths puzzle for you all to enjoy! Answers will be posted when the next puzzle is released so remember to check back and get your thinking hats on…

Below are portraits of three famous mathematicians from Ancient Greece. Your task is to give me the name of each of them along with one of their mathematical discoveries… Send your answers in on Facebook, Twitter, Instagram or via the contact form on my website. Good luck!

WARNING: answer below the picture so if you want to attempt the puzzle please scroll slowly to avoid revealing it!

Answer:

(a). Archimedes – most famous for running naked down the street exclaiming “Eureka!” after discovering what is now called Archimedes Principle. It relates the buoyancy of an object to the weight of water and allows you to easily work out whether or not something will float.

(b). Plato – involved with many things, but mathematically best known for his interest in shapes. The 5 Platonic Solids bear his name and are also my favourite shapes. Plato thought that they were so beautiful the entire universe must be built out of them…

(c). Pythagoras – perhaps the most famous mathematician to have ever lived due the triangle theorem named after him that we are all taught at school. It tells us that the length of the diagonal side of a right-angled triangle c is related to the length of the other two sides a, b by a very neat relationship a² + b² = c².

May 16, 2018 1

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.

May 9, 2018 0

How many ping pong balls would it take to lift the Titanic from the ocean floor?

The answer to the latest question sent in and voted for by YOU.

Lifting the Titanic with ping pong balls was a real suggestion put forward in the 1970’s that needless to say did not happen. Let’s pretend it is possible and work out how many we would need using Archimedes Principle…

To vote for the next question that you want answered remember to ‘like’ my Facebook page here.

October 27, 2017 1