Pi Day 2020 was the first ever UNESCO International Day of Mathematics. To celebrate we made a worldwide collaborative video on the theme ‘Mathematics is everywhere’. You can watch the full video here (I’m at 3:39) or just my contribution below – enjoy!

# 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 radius ^{3} = 4/3 x 3.14 x 2^{3} = 33.5 cm^{3}**

**Density of Atlantic Ocean = 1.027 g/cm ^{3}**

**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

# Funbers 3, Pi and 4

The fun facts about numbers that you didn’t realise you’ve always wanted to know…

**3 – THREE**

Three’s a crowd… or is it? Imagine you’re writing a story, how many lead characters would you have? One means a hero, two means a love interest, and three comes next. This is why three is so popular throughout history – it’s the first number that allows for a team, without any romance. It’s also everywhere in religion: Christianity has the Holy Trinity, of the Father, the Son and the Holy Spirit; Islam has the three holy cities of Mecca, Medina and Jerusalem; Buddhism has the three Treasures of Buddha, Darhm and Sangha; Taoism has three deities called the Three Pure Ones; Brahma, Shiva and Vishnu are the three Hindu Gods and even going a little old school there are the three Norse Norns: Urd, Verdandi and Skuld who weave the tapestry of our fate with each thread representing the life of a single person… certainly something to ponder.

**3.142… – PI**

Everyone’s favourite mathematical food and quite possibly the most famous mathematical constant. Mathematical constants are numbers that aren’t part of the usual number line and aren’t fractions, but pop up everywhere in maths. I’ve already mentioned root 2, the Golden Ratio and e in earlier articles to name but a few. Physics loves a good constant too – the gravitational constant G is a classic, and hands up if you’ve heard of h-bar? (If you haven’t don’t worry physics isn’t as cool as maths). Going back to pi, it’s of course most famous for circles. Take any circle, measure its radius (the distance from the centre to the edge) and then the area of the circle is given by pi r^{2} and the distance around the edge of the circle (also known as the circumference) is 2 pi r. The fact that this works for any circle, ever, anywhere, that has ever previously existed or ever will exist, is what makes pi such a special number. The only explanation is that it’s part of the fabric of the universe. And it doesn’t stop with circles, pi pops up everywhere in physics too. Einstein’s equations for general relativity, check. Newton’s law of gravity, check. Maxwell’s equations of electromagnetism, check. I could go on, but I’d better get onto the next number… You get the point; pi is big news.

So big in fact, that you can watch a video I made for Pi Day (March 14^{th} or 3-14 if you’re American) below.

**4 – FOUR**

Four is associated with symmetry, balance and stability. A table has four legs, lots of animals also have four legs (cows, sheep, pigs, lions, tigers, aardvarks, hippopotamuses…) and not forgetting even we have four limbs. There’s also the way we use four to divide everything up. There are four seasons in a year: winter, spring, summer and autumn, a compass has four points: North, East, South, West and there are four parts of the day: morning, afternoon, evening and night. The ancient Greeks went one step further and divided up everything in the world into one of the four elements: earth, fire, water and air. And their doctors believed that your body was filled with four liquids (which they called humors): blood, yellow bile, black bile and phlegm. If you were ill in ancient Greece, chances are your doctor would remove some of one of these liquids as an apparent cure… leeches anyone?

Then there’s the four horsemen of the apocalypse. I was a big fan of the Darksiders video game series growing up so I know all about these bad boys. They are mentioned in the Bible, though it’s difficult to know exactly what each one represents – sounds like most of the Bible to me… One is definitely death, that’s certain. The other popular choices are conquest, war (pretty similar no?) famine, pestilence and plague. Yes, that does make 6, but hey it’s the Bible, anything goes…

Oh, and I almost forgot, maths loves four too. There are four main operations: addition, subtraction, multiplication and division. It reminds me of the classic chat-up line: me plus you, subtract clothes, divide those legs and lets multiply… (crickets I know, but at least I tried).

You can find all of the funbers articles here and all of the episodes from the series with BBC Radio Cambridgeshire and BBC Radio Oxford here.

# What is Pi?

Live interview with BBC Radio Cambridgeshire for Pi Day.

# Pi Day

It’s March 14th – also known as Pi Day! Here are 3.14 fun facts about the magical number…