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
Numberphile Pi Million Outtakes
Counting outtakes from the Numberphile Pi Million Subscribers celebration video. Look out for a familiar face/torso at 4:33…
Maths with a Striptease (Die Rheinpfalz)
Tom “rocks” maths on the internet – lecturer from Oxford arouses enthusiasm with crazy ideas…
The graduate mathematician Tom Crawford not only has rock music as a hobby, but he also looks like a rock star with his tattoos and piercings. However, some of his tattoos are related to mathematics. For example, the first 100 decimal places of Euler’s number wind around his arm and the number pi has been encrypted as an infinite series. On his Youtube channel “Tom Rocks Maths” he presents science in a fun way – the clothes sometimes fly during a striptease: “I want to show that maths is not always only downright serious, but fun.”
The math lecturer from Oxford came as part of the Heidelberg Laureate Forum (HLF) in the Electoral Palatinate. Since there is no Nobel Prize in mathematics, the winners (Latin: laureates) of comparable awards are invited to the HLF. The best math and computer scientists in the world meet here for a week with junior scientists and journalists. Crawford was on the ground as a publicist and presenter, and took the opportunity to speak to some of the awardees. For example, Martin Hairer, who received the Fields Medal for his seminal studies, had an appointment for an interview. In the end, they played Tetris for an hour and talked about “cool math”: “Such a relaxed and profound conversation is only possible at the Heidelberg Laureate Forum,” the Brit enthuses about the inspiring atmosphere at the HLF.
Tom Crawford was already “packed” in the elementary school of mathematics: “When we were learning multiplication, I did not want to stop working on difficult tasks until late in the evening – it did not feel like work at all.” Even later in high school, he always did math tasks first and gladly. “I was a good student in my eleven subjects, but math was the most fun.” The satisfying thing is, “in maths a result is right or wrong, there is no need to discuss it.”
After studying in Oxford, he went to Cambridge to write his PhD in fascinating fluid dynamics. “We wanted to model how fluids move and interact with the world. I was excited about the prospect of being able to analyse experiments as a mathematician.” From this, models of reality were developed: what path does a river take when it flows into the sea? The findings help to understand the pollution of the oceans and possibly stop it. During his PhD he worked for the BBC in the science programme “The Naked Scientists”: this meant that the scientists liberated their theories from the complicated “clothes” and reduced them to a comprehensible basis. In this way, a layman will discover “naked” facts – in the sense of comprehensible ones. The radio broadcasts were a great success.”But you also have to visualize maths,” so he started to make his own videos and took the concept of the “naked mathematician” literally. In some lectures, he reveals the equations “layer by layer” and in each stage falls a garment – until Tom remains only in his boxer shorts. And then his tattoos are also visible, on whose mathematical background he will give a lecture in Oxford soon – with many guests guaranteed!
With unusual ideas, the only 29-year-old mathematician arouses the desire and curiosity for his subject. His original internet activities have now been honoured with an innovation prize. Even when attending school in Schwetzingen Tom Crawford had unusual questions: “In the stomach of a blue whale 30 kilos of plastic have been found: How much would that be if a person swallows just as much in relation to their own body weight?” The students calculated that in the human stomach, six (empty) plastic shopping bags would be located. Or, “How many table tennis balls are needed to lift the sunken Titanic off the ground?” And which example impressed him most in mathematics? “It is terrific how Maxwell’s equations, which deal first with electricity and magnetism, follow the wave property of light with the help of mathematics alone. Math is just fantastic! ”
Birgit Schillinger
The original article published in the Die Rheinpfalz newspaper (in German) is available here.
YouTube Star ‘Rocks’ Math (Schwetzingen Newspaper)
“30kg of plastic has been found in a blue whale’s stomach: how much would that be if a person swallowed just as much proportional to their own bodyweight?” Tom Crawford from Oxford began his guest lecture at Hebel-Gymnasium with this question. The students calculated that you’d find six (empty) plastic shopping bags in a human stomach. The other results worked out over the course of the entertaining presentation were also very impressive.
Tom Crawford doesn’t just have rock music as a hobby, rather with his tattoos and piercings, he looks like a rockstar too – though his tattoos are all to do with maths: since for example, the decimal places of “e” (Euler’s number) wind around his arm, the number pi is also encoded in an infinite series. On his YouTube channel “Tom rocks maths”, he presents science in an entertaining way – sometimes even pieces of clothing fly off during stripteases: “I want to show that maths isn’t always just super serious but it can also be fun.”
The mathematics lecturer is currently part of the Heidelberg Laureate Forum in Heidelberg. This is where the best maths and computer scientists in the world are meeting up with junior researchers and journalists. Crawford came to Schwetzingen at the invitation of maths teacher Birgit Schillinger. He brought along some exciting questions. The common theme was Tom’s favourite number, pi, which is used in so many formulas. How many ping pong balls are needed to lift the sunken Titanic off the ground? Which factors are involved when a footballer bends a ball so that it flies in an arc past the wall into the goal? When calculating the trajectory, several physical variables play a role. But how? Crawford studied the mathematics behind it. His doctoral thesis was on fluid mechanics: What path does a river take when it flows into the sea? The findings help us to understand sea pollution and possibly help to stop it.
At the end, the Hebelians made Platonic solids, of which, amazingly, there are only five. Strange? No, Tom explains this number by the sum of the angles at the corners – all very logical! Finally a student’s question, which example in mathematics has impressed Tom the most: “It is terrific how the wave characteristic of light follows from Maxwell’s equations, which deal with electricity and magnetism, with only the help of mathematics. Maths is just awesome!”
Birgit Schillinger
Thanks to Cameron Bunney for the translation.
The original article in Schwetzingen can be found here.
Funbers 25, 26 and 27
Fun facts about numbers that you didn’t realise you’ve secretly always wanted to know…
25 – TWENTY-FIVE
You probably know 25 as five squared, 5 x 5 = 25, but I bet you didn’t realise that it’s also the sum of the first five odd numbers: 1 + 3 + 5 + 7 + 9 = 25. It also crops up a lot in Pythagoras’ Theorem (yes, him again — see Funbers root 2) because it’s the smallest square that’s also the sum of another two square numbers: 25 = 3² + 4². Since Pythagoras’ Theorem says that a² + b² = c², we have the exact result with whole numbers (integers) for a = 3, b = 4 and c = 5. A solution such as this, where all of the numbers are integers, is called a Pythagorean Triple.
Looking beyond the maths, most videos are recorded at a frame rate of 25 per second as the PAL video standard – other options are available, but twenty-five does an excellent job of tricking the human brain into seeing a moving picture where in fact only a series of still images are being shown. Less than 25 and we might start to notice the ‘jumps’ between frames, and for more than 25 we’ll need a lot more data to record and store the footage.
Twenty-five is also the average percentage of DNA overlap between yourself and your grandparent, grand-child, aunt, uncle, nephew, half-sibling, double cousin (when siblings from one family have children with siblings from another), or identical twin cousin (if one of your parents is an identical twin and their twin has a child). Oh, and apparently a ‘pony’ is British slang for £25 – news to me…
26 – TWENTY-SIX
With twenty-five being a square number, and (spoiler alert) twenty-seven being a cube number, twenty-six is uniquely placed as the only whole number that’s exactly one greater than a square (5² + 1) and one less than a cube (3³ – 1). Talk about niche. And then there’s the fantastically named rhombicuboctahedron — a shape with 26 faces, made up of squares and triangles. Can you spot how many of each in the figure below?
Twenty-six also gives the number of complete miles in a marathon (26 miles and 385 yards to be exact), the number of letters in the Latin alphabet, and the age at which males can no longer be drafted in the United States. The draft has been used five times throughout history: the American Revolution, the American Civil War, World War 1, World War 2 and the Cold War (including Korea and Vietnam). Let’s hope it never has to be used again.
27 – TWENTY-SEVEN
Now this one’s a real doozy: 27% of our universe is made up of “dark matter” – matter that has mass but is also completely invisible and doesn’t interact with itself or regular matter. The rest of the universe consists of 5% regular matter (the stuff we know about), and the other 68% is completely unknown. Something, something, dark energy…
Sticking with scary thoughts, in Stephen King’s novel ‘It’ (great film by the way) the creature returns to the town of Derry every 27 years, which also happens to be exactly the right amount of time for a new-born baby to join the 27 Club — a term used to refer to popular musicians who have died at the age of 27. Current members include Jimi Hendrix, Kurt Cobain and Amy Winehouse amongst many, many more. We also have 27 books in the New Testament and 27 bones in the human hand.
Ending with some maths — what else — twenty-seven is the only positive whole number that is exactly three times the sum of its digits: 2 + 7 = 9 and 9 x 3 = 27. It’s also a perfect cube, 33 = 3 x 3 x 3 = 27, and it’s equal to the sum of the digits from two to seven, 2 + 3 + 4 + 5 + 6 + 7 = 27. But, leaving the best until last, if you label the decimal places of the number pi, starting from 0, then the 27th and 28th digits read 27. It may seem like magic but it’s actually one of a few ‘self-locating strings’ in the number. The others being 6, 13598, 43611, 24643510, and no doubt many more yet to be discovered. That can be your homework…
π = 3.141592653589793238462643383279…
Image credit: Jonathan Kis-Lev
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 10^{23}.
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 |
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.