Make your own Pi

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

March 14th 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 14th 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 31st April, which unfortunately isn’t a real date in the Gregorian calendar. So, March 14th 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

quantum

Relativity: Einstein Field Equations

relativity

Gravity: Newton’s Gravitational Constant

gravity

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.

circumference

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:

volume

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:

Screenshot 2020-03-16 at 13.00.29

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):

Screenshot 2020-03-16 at 13.00.29

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

Residencia de Estudiantes, Madrid

This week I had the honour of speaking at the Residencia de Estudiantes in Madrid, which has previously hosted Albert Einstein, Marie Curie, Salvador Dali and Igor Stravinsky amongst many, many others.

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Ahead of the event I was asked a few questions by the organisers, and here are my answers.

Without revealing all your talk: could you give us an idea about how maths can help to be better at sports? 

From calculating the perfect placement of a penalty kick in football to maximise your chance of scoring, to identifying the best location on Earth to try to break a world record, maths can be used to help to improve our performance in almost any sport. The difficultly lies in writing down the correct equations, but once we have them, maths has the answers.

Can you tell us any real example of this maths application?

My favourite example is one that will be featured in my talk: if attempting to break a world record in rowing, the best place to do so is on the equator. This may seem counter-intuitive at first, but as I will explain, by changing the location to the equator you can increase performance by up to 8%, which for an elite athlete is an incredible boost!

In your opinion: what makes maths so useful in different sports context?

Maths can be applied to anything. This is one of the main reasons that I love the subject and travel the world championing its versatility. Given a situation in any sport, you can always use equations to describe what is happening. This might be how a tennis ball moves through the air, or the aerodynamics of a swimmer gliding through the water. Once you have the equations, maths allows you to solve them for the optimal solution, which can then be translated into improved performance by changing your technique appropriately.

You also explain that the mathematical results in sports may vary, how?  In which way? What should athletes take into account?

The ideas discussed in my talk are aimed at professional athletes who are already performing at a very high level and therefore need to resort to other approaches to improve performance beyond increased practice. For amateur athletes, whilst the same ideas will still be applicable, they are much more likely to benefit from practice!

What is your personal experience with sports? Have you ever used “math tricks” for optimise your scores?

The idea for the talk came from wanting to combine my two main passions: mathematics and sport. I play football regularly and as the designated penalty taker for my team have ample opportunity to try to hit the mathematically calculated perfect position for a shot. I also run marathons where my knowledge of the history (and mathematically predicted future) of the world record helps me to appreciate my accomplishments in the event.

IMG_20191112_145340

How did you become a math communicator? 

My first taste of maths communication came during my undergraduate degree at Oxford, where I joined the maths outreach group “Marcus’ Marvellous Mathemagicians”. The group was named after Marcus du Sautoy and performed interactive talks and workshops on his behalf in schools across the UK. The next opportunity came during my PhD when I spent two months working with the “Naked Scientists” team in Cambridge to produce a weekly science radio programme for the BBC. I enjoyed the placement so much that I agreed to join the team full-time upon completion of my PhD. After one year of working in radio production, I began to realise that my true calling was in video, and “Tom Rocks Maths” was born.

How are outreach, teaching and research connected in your professional life?

As someone who came from a state school background and worked extremely hard to get to Oxford, I have always had a passion for outreach and the drive to make university accessible to all. My maths communication work is an extension of this, allowing me to not only to visit schools in deprived areas to try to inspire them to consider higher education, but also to encourage the general public to engage more with the subject of maths and to no longer be afraid of numbers.

The teaching role fits perfectly with maths communication as both roles require the ability to be able to explain difficult concepts in ways that can be understood by a given audience. For a public lecture, the mathematical ability of the audience is perhaps less than that of a class of undergraduates, but the need for clear communication remains the same. In this way, I find that each role complements the other perfectly, with many of the topics that my students find difficult providing inspiration for future video ideas.

What do you enjoy most in your outreach talks? 

There is nothing I enjoy more than being able to present to a live audience. Whilst I enjoy all aspects of my outreach work – YouTube, television, radio, writing – nothing beats the thrill of speaking to a room full of people who want to hear what you have to say. The small interactions with each individual member of the audience, whether through eye contact or answering a question, remain with me long after the event and act as one of my main motivations to continue with my work.

You are not the speaker one might expect when thinking about a maths communicator, what kind of reactions have you find in this sense? Do you have any anecdote regarding this? 

There are two ways of looking at this: first, the notion of a stereotypical mathematician is outdated and from my experience not representative of a large part of the demographic; and second, I hope that by putting myself forward as a public face of mathematics I can help others who may be thinking that they can’t be a mathematician just because of the way that they look.

In terms of anecdotes, I think it best that I point you in the direction of the comments on my YouTube videos…

In particular, what are the reactions with “Equations stripped”? How did you come up with the idea of this series?

The “Equations Stripped” is possibly my favourite of all of the things that I do because it helps to tackle the idea that maths should be serious. The concept of the videos came from thinking about this opinion and trying to come up with what I thought was the best way to present the subject as anything but serious. The result is me talking about maths in my underwear!

My role with the “Naked Scientists” also played a part, as the name would often lead to listeners (or even guests) suggesting that we should all be naked when recording the show, and of course being a radio programme no-one could prove or disprove the theory! I always thought that we should have had more fun with this concept, and when “Tom Rocks Maths” was launched Naked Maths seemed like the way to go!

Tom Rocks Maths S02 E11

A special edition of Tom Rocks Maths on Oxide Radio with music inspired by Tom’s recent visit to Slam Dunk Festival. We’ve also got Pokemon and drinking games, a mind-bending Einstein Puzzle, and news of Tom’s antics running around the streets of Oxford in his underwear… This is maths, but not as you know it.

The Heroes of Sir Michael Atiyah

In the final part of my interview with Sir Michael Atiyah – one of his last ever before he passed away – he talks about some of his mathematical heroes, from Einstein and Newton to Brouwer and Michelangelo, including the most beautiful description of the ceiling of the Sistine Chapel I’ve ever heard. A true giant of Mathematics, who is sorely missed.

With thanks to the Heidelberg Laureate Forum.

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