Tag: pi

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.”

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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.

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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.

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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?

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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.

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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

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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!

Catch22-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…

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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.

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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

 

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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.

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Funbers 2 Pi

Two times pi is such an important number in maths that it deserves its very own edition of Funbers… featuring angles, trigonometry and pie jokes.

You can listen to all of the Funbers episodes from BBC Radio Cambridgeshire here.

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What is the Gamma Function?

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

We’ve got probability distributions, complex analysis and of course Pi (because it appears everywhere)… I give you the Gamma Function.

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

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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 r2 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 14th 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.

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