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

Teaching Mathematics

Following my talk in Madrid in November, I was asked to answer a few questions about the current status of maths teaching based on my experience as a university lecturer. Here are my answers…

How should mathematics be taught in schools?

Through stories. Teaching through story-telling is an incredibly powerful tool and one that is not used enough in mathematics. For example, when teaching trigonometry, rather than just stating the formulae, why not explain WHY they were needed in the first place – by ancient architects trying to construct monuments, by explorers trying to estimate the height of a distant mountain – these are the reasons that mathematics was developed, and I think that teaching it through these stories will help to engage more students with the subject.

Are teachers prepared to teach this subject correctly?

I don’t believe the teachers are at fault – they are told to follow a particular curriculum and due to their heavy workload have no time to develop lessons with engagement at the heart of their design. There are of course ways that we can help teachers, by providing examples of ways to make maths content more interesting and engaging. This can be through story-telling or applications to topics of interest to students such as sport and video games. This is what I try to do with ‘Tom Rocks Maths’, for example see my video teaching Archimedes Principle by answering the question ‘how many ping-pong balls would it take to raise the Titanic from the ocean floor?’.

In your view, how should a math teacher be?

The most important thing is to have passion for the subject. The level of excitement and interest that the teacher demonstrates when presenting a subject will pass on to the students. Just as enthusiasm is infectious, so too is a lack of it. Beyond passion, there is no typical profile of a maths teacher. Anyone can be a mathematician, and it is very important that people don’t feel that they have to conform to a particular stereotype to teach the subject. I have always just been myself, and hopefully as a public figure in mathematics will inspire others to do the same.

Sometimes, this subject becomes more complicated for some students, not so much because of its difficulty, but because of the way in which they have been taught. What should be done with these students?

The trick is to find a way to explain a topic that resonates with a particular group of students. Let me give you an example from my research: the Navier-Stokes Equations (NSEs). For students who have no real interest in mathematics, I would try to get them to engage by explain the $1-million prize that can be won by solving these equations. For students who have more interest in real-world applications such as in Engineering or Biology, I would tell them about how the aerodynamics of a vehicle or the delivery of a drug in the bloodstream rely on an understanding of Fluid Mechanics and the NSEs. If the students are fans of sport, I can explain how the equations are used to explain the movement of a tennis ball through the air, or for testing the perfect formation in road cycling. Finally, for students who are already keen mathematicians, I would explain how the equations work in almost every situation, except for a few extreme cases where they result in ‘singularities’, which as a mathematician are the ones you are most interested in understanding. Once you know the interests of your audience, you can present a topic in a way that will help them to engage with the material.

Can you get to hate math?

It is certainly possible – though of course alien to mathematician such as myself! I think this feeling of ‘hate’ relates back to either the way that you have been taught the subject, or from a lack of understanding. If you did not enjoy your maths lessons at school and harbour ill feelings towards your teacher, then you will begin to develop negative feelings towards the subject. This is not because you dislike the subject, but more because of the way that it was taught to you. Likewise, if you do not understand mathematics then it is very easy to develop a ‘fear’ of the subject, which can quickly turn into hatred due to feelings of inadequacy or stupidity if not addressed. It all comes back to finding a way to approach the subject that fits with the knowledge and experiences that you already have. If you present a problem in an abstract manner of manipulating random numbers to find a given total, then most people will struggle – regardless of their mathematical ability. But the same problem presented in a relatable situation suddenly becomes understandable. Here’s an example:

(a). Using the following numbers make a total of 314: 1, 1, 2, 5, 10, 10, 20, 20, 50, 100, 100, 500.

(b). You go shopping and the total is €3.14. What coins would you use to pay for your items?

They are the same question, but in (a). the problem looks like a maths question, and in (b). it is an everyday situation that people all over the world are used to. Both require the same maths to solve, but even people who ‘hate’ maths could tell you the correct answer to (b). using their own real-life experience.

Women are at a great disadvantage compared to men when entering a STEM career, why do you think this is happening?

First of all, as a man I am certainly not qualified to answer this question, but I will at least try to provide you with my opinion based on personal experience. At high school level I believe that the difference is less severe (eg. see article here) and even at university there is a slightly higher number of females than males studying science-based subjects. BUT, the issue occurs after this. In graduate degree programmes and beyond there is a definite lack of female researchers, and this is amplified even further at more senior level positions. One explanation could be that academic ‘tenure-track’ positions exist for life, and so many of the men that now hold these positions have done so for the past 30-40 years and were employed when we were doing a much worse job of tackling the gender gap. Now that awareness of these issues has increased, and in general we are doing a much better job at addressing them that we were 30 years ago, hopefully we will begin to see more females in leading positions over the coming years, it will just take a little while for the effect to be seen. I also think that in general there are not enough female role models within many subjects (especially maths) that have reached the pinnacle of their field (through no fault of their own), and as such there is a lack of role models for young female researchers. The achievements of female mathematicians such as Maryam Mirzakhani (2014 Fields Medal) and Karen Uhlenbeck (2019 Abel Prize) should be even more celebrated precisely for this reason.

Do you think that enough importance is given to mathematics in the educational world?

In the past perhaps not, but attitudes are certainly changing. With the increased role that technology and algorithms play in our lives, people are beginning to realise that we need to better understand these processes to be able to make informed decisions – and maths is the key to doing this. Employers are certainly aware of the invaluable skillset possessed by a mathematician and as a result more and more students are choosing to study the subject at degree level and beyond to improve their competitiveness in the job market. Ultimately, attitudes are changing for the better, but there is still more that can be done.

In your opinion, what is the best way to teach this subject?

Exactly as I have described in questions 1 and 4. Storytelling is key to making the material as engaging as possible and knowing the interests of your audience allows you to present the subject in a way that will appeal to them most effectively.

What is the current situation of mathematics research in the university?

I think the main issue facing research mathematics is the relatively recent trend of short-term research outcomes. The majority of funding available to mathematicians requires either continuous publication of new results or outcomes that can readily be used in an applied setting.  The issue of continuous publication means that researchers feel the need to publish a new manuscript every few months, which leads to very small advances at each step, and a wealth of time spent writing and formatting an article instead of conducting actual research. In many cases, the work would be much clearer if published as one piece in its entirety after several years of careful work. The drive for short-term research outcomes means that it is now very difficult to study mathematics just for the sake of it – you have to be able to convince your funding body that your work has real-world applications that will be of benefit to society within the next 5-10 years. To show why this is a disaster for maths research, let’s take the example of Einstein and his work on relativity. Now seen as a one of the most fundamental theories of physics, his work had no practical applications until the invention of GPS 60 years later. In today’s short-term outcomes driven market, it is highly unlikely that Einstein’s work would have been funded.

Photo: Residencia de Estudiantes

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