The Valknut Challenge

I’ve been lacking in time and inspiration for blogging for a while now, but hopefully will get back into it again now that some of the chaos of Semester 2 has passed. As my come-back article I wanted to write about a fantastic evening I had at the RBS Museum Lates at the National Museum of Scotland just over a week ago.

The Museum Lates happen three times a year and are a chance for adults to get into the museum after hours to look at the collections whilst enjoying a cocktail and some live music. But they are much more than that! Each Late is themed in some way, and the theme this month was ‘Vikings’ to accompany the museum’s special exhibit. So guests were encouraged to dress up as Vikings, get their faces painted, touch Viking objects, make Viking paraphernalia (but NOT horned helmets – they aren’t Viking!), listen to Viking stories and eat Viking food. It was the perfect opportunity for us to get in some stealth maths engagement…

(From left) Joshua, Helene and Madeleine all Vikinged up and ready to go.

My co-conspirator in this project was Madeleine Shepherd (who you may remember from lots of previous projects, including the last Alice In Wonderland themed museum late and also the Mathematicians’ Shirts project), and on the night we had two lovely assistants: Helene Frossling (Madeleine’s colleague at ICMS) and Joshua Prettyman (an undergraduate on my maths outreach team). Together we presented the public with…The Valknut Challenge!

In Viking mythology, the Valknut was the special symbol of Odin, king of the gods. It consisted of three interlocking circles, but if you removed any one of the circles then the other two would fall apart and would not be linked together at all. The Valknut Challenge is: can you draw/make the symbol? If you haven’t seen this before, you should have a go before scrolling down to see the picture.

The Valknut symbol is often found on Viking stones where Odin is depicted going into battle or standing over fallen warriors. It’s not hard to see why the symbol would be synonymous with strength in battle: the three interlocking circles symbolise the strength we have in acting together which falls apart when we go it alone. The same symbol has been found in many civilisations – for example, signifying the Trinity in Christianity.

The Stora Hammars stone showing a Valknut above a scene of human sacrifice and next to some warriors. Powerful stuff.

Mathematically the symbol is called the Borromean rings (named for the Italian Borromeo family who used the symbol on their coat of arms). There are actually infinitely many different ways of solving the puzzle, and the Valknut is a special case of a more general puzzle where you have n circles and have to interlink them so that removing one ring makes the rest fall apart. Such a link is called a Brunnian link and they are particularly cool topological objects.

The standard Borromean rings.

A different solution to the Valknut challenge.

And another solution!

Most people have created a Valknut/Brunnian link in the course of their lives without ever realising it. Every knot or link can be drawn as a braid, and the Valknut is actually the standard 3-stranded braid that girls do in their hair all the time. Try making one and pulling out a strand – you’ll find that the other two strands become instantly untangled. This means you can quickly make a braid and harness the power of Odin whenever you need it – handy to remember next time you find yourself in battle.

The next Museum Late will be on 17th May with the theme of ‘Dinosaurs’.  If you missed out this time around then make sure you get tickets early for the next one! Anyone have any ideas for some surreptitious Jurassic mathematics that we can do?

My new geek clock

I’m loving this new addition to my flat: a geeky maths/science clock. It was designed by myself and Albert and then built by Kate Wharmby, who is also very talented at making stained glass and has made me a lovely trefoil knot too.

I thought I would go through the numbers and say what they mean and why we chose them. All physical constants are in standard SI units.

1. ℏ × 10³⁴ = 1.05 is the reduced Planck constant; that is, the Planck constant divided by 2π. This number relates the energy of a photon to its angular velocity and is one of the basic units in quantum mechanics. Its existence means that energy only comes in discrete packets, or quanta, which was a massive discovery in 20th century physics.
2. is the sum which turns out to be exactly equal to 2. It’s an example of a geometric series and was first discussed in ancient Greece when Zeno came up with his paradoxes. He claimed that motion was impossible, because before you could get somewhere you’d have to get halfway there, but then you’d have to get halfway to the halfway point and so on. However, despite the infinite number of steps in the sequence it actually has a finite value.
3. c × 10^-8 =2.9979 is the speed of light in a vacuum, measured in metres per second. The theory of relativity says that this is constant, regardless of the observer, and that nothing in the universe can go faster than this. The fact that c is constant leads to lots of crazy consequences, such as time passing slower for moving objects and moving objects shortening in length.
4. is the integral (area under the curve) of the natural logarithm function between and . Logarithms were invented by the Scottish mathematician John Napier and are important in many branches of maths and science. The natural logarithm is related to the mathematical constant e, which arises whenever a quantity is growing at a rate proportional to its current value; for example, compound interest.
5. det(12n₄₈₈) is the determinant of the 488th non-alternating 12-crossing knot. This was a knot which featured in my thesis, as I used new techniques to show that it had infinite order in the knot concordance group. Finding the determinant was the first step in this method. You can find tables of knotty things on the website KnotInfo, including any values which are still unknown.
6. 29iⁱ = 6.03 shows off the surprising fact that an imaginary number (the square root of -1) to the power of an imaginary number is actually a real number. Actually, iⁱ  is equal to .
7. is the golden ratio squared and then squared again. The golden ratio arises if you want to divide a line into two parts so that the ratio of the larger to the smaller part is the same as the ratio of the larger part to the whole line. It is often described as the most aesthetically pleasing ratio, and is also the ‘most irrational’ number in the sense of being most badly approximated by rational numbers.
8. N_A.k_B = 8.314 is the ideal gas constant which relates the energy of a mole of particles to its temperature. It is the product of N_A, which is Avogadro’s constant (the number of particles in one mole of a substance – see 12), and k_B, which is the Boltzmann constant (which relates energy of individual particles to temperature).
9. m_e × 10³¹ = 9.109 is the rest mass of an electron which is one of the fundamental constants of physics and chemistry. It is actually impossible to weigh a stationary electron, but we can use special relativity to correct our measurements of the mass of a moving electron.
10. g = 9.81 is the acceleration of objects in a vacuum at sea level due to Earth’s gravity. This is actually an average value, as the force of gravity varies at different points on the Earth’s surface. Things affecting the apparent or actual strength of Earth’s gravity include latitude (since the Earth is not a perfect sphere) and the local geology (since the Earth does not have uniform density). The force is weakest in Mexico City and strongest in Oslo.
11. = 11.23 is a combination of everyone’s favourite mathematical constants, pi and e. These numbers are not only irrational but are transcendental, meaning that they are not the roots of any integer polynomial equation. It is still unknown if is transcendental, although we know that is.
12. ¹²C = 12 is the isotope of carbon having six protons and six neutrons in its nucleus, giving it a mass of 12 atomic mass units. More importantly for chemists, IUPAC defines the mole  as being the amount of a substance containing the same number of  atoms (or molecules, electron, ions etc) as there are the number of atoms in 12g of carbon-12 (which is Avogadro’s number – see 8).

Which one was your favourite number? If you were designing your own geek clock, what numbers would you pick? Leave your answers in the comments!

Guest Post: Topological Crystallography in Stockholm

Here I am at one of the beamlines at Petra synchrotron, at DESY, Hamburg. The tube behind me is where the beam comes from… scary!

Albert here! Some of you may recognise me from Haggis’ Twitter feed and from Haggis’ 2011 New Year’s post (along with the rest of our family!). Last week I was in Hamburg at PETRA III, a synchrotron at DESY. After some successful measurements there, I made the short hop across the Baltic Sea to the lovely city of Stockholm, for the 4^th International School on Crystal Topology.

First I should say a little about what I do. I’m interested in chemistry, especially materials called Metal-Organic Frameworks (MOFs).

An example of one of the first MOFs, MOF-5. Chemists use rigid organic struts (top left) to link clusters of metal atoms (in this case four zinc atoms, bottom left) to build open framework-like materials (right).

These are a new type of material made from clusters of metal and oxygen atoms which are linked together by long rigid linkers – think of it kind of like a climbing frame. These materials are interesting as they might help to combat climate change by sieving out CO₂ in a process called Carbon Capture and Storage (CCS made it into the Oxford English Dictionary recently!).

But what does this have to do with topology? Chemists simplify the structures of MOFs down to a series of rods (edges) and nodes where these rods meet (vertices) – the simplified structures are mathematical graphs. We can then see how the structure is connected together as a network, without unnecessary molecular clutter. As chemists we want a way to classify the networks of our materials for two reasons. Firstly, so we can see if similar networks have been made before by other researchers, and secondly to help us design new materials. We might, for example, find that a certain network is really good at storing CO₂; using a linker molecule which holds onto CO₂ really well and the right topology to form our target network, we could make a new material which is even better at capturing CO₂. To classify our networks we need to use graph theory.

Charlotte Bonneau (left), Michael O’Keeffe (middle left), the person I hitched a lift to Stockholm with (middle right), Xiaodong Zou (right)

However chemists are not normally trained in graph theory, so this was the aim of the Stockholm school. The school was taught by Prof. Michael O’Keeffe (emeritus Regents’ Professor at Arizona State University), who taught us about the mathematical ideas necessary to deconstruct a crystalline network, and Dr Charlotte Bonneau (currently a full time mother to the adorable Leonie), who focussed more on the use of software to analyse crystal structures, such as systre and Topos.

During Mike’s lectures we were told about the graph isomorphism problem of determining whether two finite graphs have the same connectivity. This is of importance to chemists, as we want to be able to compare our networks to see if they have been reported before! Graph isomorphism is also a specific example of one of the million dollar maths problems, P versus NP, which asks whether every problem for which a solution can be quickly checked, may also be quickly solved by a computer. One of Mike’s collaborators, Dr Olaf Delgado-Friedrichs, has attempted to address the graph isomorphism problem in the program systre. systre uses a barycentric method to raise the symmetry of a collection of atoms in a graph to the highest symmetry representation. The barycentric representation is effectively like replacing all the edges in the graph with springs and these pulling the vertices to their weighted average positions. Although systre is able to classify most graphs, it is unable to deal with graphs where applying the barycentric approach causes two nodes to collapse into one another (a so-called collision – see picture). So unfortunately, it’s not a complete solution to P versus NP.

A graph showing a collision. When you put this into a baricentric representation, the two red nodes collapse into one another. Back to the drawing board for a solution to the graph isomorphism problem then…

The rest of the course was full of lots of useful information which will help in making new materials and further classifying old ones. The course as a whole was a lot of fun and it was great to meet such a friendly bunch of people! That’s it from me for the minute, but look out for more photos of me on Twitter at exciting scientific/mathematical locations – Albert out.

A Night in Wonderland

On 18th May I was lucky enough to get involved with my first RBS Museum Lates at the National Museum of Scotland. These events happen about 3 times a year and are a chance for the (over 18) public to come back into the museum after hours and to get cosy with the exhibits with a cocktail and live band. It’s also a chance for science (and arts!) communicators like me to run an activity and get some surreptitious education into the evening.

The theme for this month’s Museum Late was “A Night in Wonderland”, so there were lots of top hats, white rabbits and red queens! (See lots of photos of the event on the Museum’s Flickr page.) Knowing that Lewis Carroll (real name Charles Dodgson) was a mathematician and logician as well as nonsense-poem writer, it seemed wrong for there not to be a mathematical component to the evening so I got together with Madeleine Shepherd (from ICMS) to brainstorm some ideas…

Our first idea was to get the public to make some Fortunatus’ purses. A Fortunatus’ purse appears in the novel Sylvie and Bruno by Lewis Carroll and is based on the old tale of Fortunatus, who has a purse which replenishes itself with money as often as coins are drawn from it. If you read the book you’ll find instructions for making such a purse by sewing together the edges of 3 handkerchiefs in an unexpected way.

‘You shall first,’ said Mein Herr, possessing himself of two of the handkerchiefs, spreading one upon the other, and holding them up by two corners, ‘you shall first join together these upper corners, the right to the right, the left to the left; and the opening between them shall be the mouth of the Purse.’

Now, this third handkerchief,’ Mein Herr proceeded, ‘has four edges, which you can trace continuously round and round: all you need do is to join its four edges to the four edges of the opening…’

The mathematical object created is one which has no inside or outside – it is called non-orientable, and is (of course) not possible to make in 3 dimensions without part of the purse intersecting itself. Some of you may be thinking that this is a Klein Bottle, but it is actually a different creature called a Projective Plane.

However, whilst doing the practice run for the purse-making, we found that it took quite a long time, was fairly fiddly and would involve giving drunk people sharp needles. Probably not the best idea. (But we might do this in a future maths/craft event!)

So instead we came up with the “Snark Constellation Challenge”, inspired in equal parts by the Lewis Carroll poem The Hunting of the Snark and by a mathematical object in graph theory called a snark. Visitors were invited to play a game which involved colouring the lines between stars in a constellation, and were challenged to colour the lines using only 3 colours.

Can you colour the lines with 3 colours so that at each star, 3 different colours meet?

There were two games the visitors could play: working collaboratively to find a colouring of all the lines, or working competitively to be the last person to draw a valid line. Have a go at the puzzle and see if you can colour the lines before reading on!

Edinburgh Sci Fest 2012 (Part 2)

Welcome back to part 2 of my write up of our exhibit at the Edinburgh International Science Festival. As you may remember, we were running a series of games and activities to test people’s probability skills and to see how people would react to the stats in a courtroom. In this post I will go through the solutions to the various questions we asked, so if you haven’t had a go at them yet then make sure to have a go now!

Edinburgh Science Festival 2012

Hello maths fans!  It’s been a very busy semester for your favourite geek sheep: sorting out activities for undergrads in Innovative Learning Week, lecturing Y1 undergrads in Proofs & Problem Solving, organising business/academic networking events, doing an art/science exhibition, and running an exhibition at the Edinburgh International Science Festival. Hopefully now that I have some time, you can look forward to blog posts about all of these things.

Today’s post is about our science festival fun. We (the School of Maths) teamed up with the School of Chemistry and went for a CSI-themed activity.The premise was that a priceless Egyptian vase had been stolen from the Museum and the visitors had to work out whodunnit. Using chemistry they had to analyse fingerprints and blood samples, and use UV and infrared data to identify substances found at the scene of the crime. After deciding on their prime suspect, they came over to the maths section, which was the courtroom. Here they had to weigh up the probabilities and statistics and then decide on whether their suspect was innocent or guilty.

Just as in real life, we didn’t reveal who actually did it, because we often don’t know for sure. And actually, we hoped that (despite all the evidence) the visitors would vote ‘Innocent’ because the evidence certainly didn’t prove anything beyond all reasonable doubt.

Most £20 notes have traces of cocaine on them.

I’ve had my heart set on doing something like this for a while because I wanted to publicise the great work that our Forensic Statisticians (Colin Aitken and Amy Wilson) are doing right here in Edinburgh. They are analysing the occurence of drugs like cocaine on banknotes to help the police decide when someone is really a drug dealer. Apparently (and don’t quote me on this) most £20 notes (like, over 80%) have got traces of cocaine on them, so the police need help in deciding when the notes have been involved in drugs crime or when they have just accidently been placed next to the ‘dirty’ notes in a shop till.

Colin has also appeared as an expert witness in a few trials and has helped to write books to educate judges and lawyers about statistics. Like the general population, judges and lawyers often have a very bad intuition about probabilities. But unlike the general public, their decisions can really affect people’s lives. The classic example is the Sally Clark case. An expert witness for the prosecution claimed that there was a 1 in 73 million chance that two cot deaths could happen naturally in the same family, and therefore that Sally must have murdered her children. Not only was this statistic wildly wrong (the actual figure is about 1 in 100,000) but the conclusion of guilt is also wrong. Neither side took into account the probability of her innocence: despite the unlikelihood of double cot death, double murder is (statistically) even more unlikely. Such a mistake is called the Prosecutor’s Fallacy. In Sally’s case, it led to her spending 3 years in prison for a crime she never committed and then committing suicide a few years after she was freed.

So anyway, the idea behind our science festival exhibit was to show people how bad they were at judging probabilities and to introduce the idea of Bayesian Statistics (which is behind things like the Prosecutor’s Fallacy). Have a go at these questions and see if you can solve them! Answers will be provided in the next blog post.

The Monty Hall problem assumes you'd rather win a car than a goat. This is not true for everybody.

1) One of the most famous examples of conditional probability is called the Monty Hall problem, or the Car-Goat problem. You are on a gameshow, trying to win a car. You definitely don’t want to win a goat. There are three doors, behind which the host of the show has hidden 2 goats and a car. You choose the door which you think conceals the car. The host then opens a different door to reveal a goat. Finally, you get to choose: should you stick with your original choice of door, or should you swap? Or does it make no difference?

2) On very similar lines is the following queston. I flip two coins and tell you at least one of them is a head. What’s the chance that the other one is also showing a head?

3) In a lottery there are a 10 numbers in a bag and you win the jackpot if you correctly predict which 4 numbers get pulled out. What are the chances of winning the jackpot? What are the chances of predicting 3 out of the 4 numbers?

4) If 100 people each flip a fair coin 5 times, how many of them will we expect to flip 5 heads?

5) On the wall there is a calendar for 2012. Visitors to the museum put their birthday on the chart as they come in. After how many visitors do we expect to see the first shared birthday? (I.e. two people with the same birthday.)

Betty only sees what she thinks 2/3 of the time.

6) An eyewitness, Betty, says she saw a suspect leaving the scene of the crime, and that the suspect was wearing a hat. Betty is shortsighted and only correctly identifies hats 2 out of 3 times. That is, 1 time out of 3 she will think that someone is wearing a hat when they aren’t, and 1 time out of 3 she will think that someone isn’t wearing a hat when they are. If 10% of the Edinburgh population wears a hat, what are the chances that the suspect was really wearing a hat?

Needless to say, most visitors to the exhibit found these questions very difficult, but that was the point. We wanted to teach people not to trust their intuition when it comes to probability, and especially not if they are in a jury on a court case!

Many thanks to all who visited us in the Museum and played all our games with us! I hope you all had a good time and learnt something new.

It’s not quantum physics

As we finish the Edinburgh & Lothians maths masterclasses (which have been a great success!) I am still following the efforts of another group of young people – those on The Young Apprentice.

The Young Apprentice candidates

These are apparently the best 12 young entrepreneurs of the country, chosen from many thousands to compete for Alan Sugar’s money to set up a new business of their own. Initially they were divided into two teams (boys and girls) and asked to choose names for their teams. Independently, both teams went for science-sounding names: Atomic and Kinetic. This bodes well, I thought. At the very least, it shows that young people view science as exciting and fast-paced; at best, maybe these future entrepreneurs have a deep appreciation for maths and science.

Ha! Wishful thinking.

Despite the Physics-sounding team names, only 2 of the 12 contestants are studying any kind of science, and in both cases it is Biology. 3 of them are doing an A-Level in Maths and a few more in Economics, but we’ll see soon enough that this implies nothing about their basic arithmetical skills.

Episode 1: The contestants have to make and sell their own brand of ice cream. Problems start early on, when they have to work out how many kilograms of ingredients are necessary to make so many litres of ice cream. Not just that, but they have to work out, given how many scoops they can sell in an hour, how many litres to make in the first place, and how much profit will accrue from various costings. The voice-over dubs this ‘A-Level Maths’. Hmm. Is multiplication and division really not taught before the age of 16 any more?

Sainsburys clearly need to employ someone with an A-Level in Maths in order to do their pricing...

But maybe the voice-over man is right. The boys have a team member with A-level maths, who copes fine with the numbers. The girls, on the other hand, have sent both their A-levellists off to do the design work, leaving chaos behind them. The remaining team think a gram is equivalent in weight to a litre. They think 3 x 4 is 28. “The real surprise here is that the team cannot add up, subtract, divide or multiply”, says a bewildered Nick Hewer, who is keeping an eye on proceedings. In the boardroom, the project leader defends herself saying “I only have a GCSE in maths”, to which Lord Sugar responds with “I don’t care if you only have air miles in maths – this is baby stuff!”

Episode 2: The contestants have to design a new product for the mother & baby market. Thankfully it’s more about market research and good pitching than making and selling this week, so no maths to worry about.

Episode 3: The contestants have to set up their own floristry businesses, designing and selling displays to corporate customers, as well as flogging bouquets on the street. Each team is divided into two, with some people staying behind to learn flower arranging and the others going to win deals with companies. Maths makes an early appearance, with one team leader choosing to keep one of the girls in the flower arranging group, despite her being a proven success at pitching to companies. The reason? Because she’s the only one who can do the maths in pricing the displays.

One of the boys, who had got the highest score in GCSE Economics in Northern Ireland, proclaimed “I’m not very good with numbers.” Prompted further, he said “There’s not a lot of numbers in Economics”, to which Lord Sugar replied “We’re not talking about quantum physics here, are we? We’re talking ‘this rose costs 40p, so 10 roses costs £4′.”

In the end though, the most frustrating thing for me about watching the show is that the team who gets it right every week is almost certain to lose the task! The teams who work hard to correctly do their costings, to make a good product and to provide good customer service always fall down because they haven’t overcharged for their product like the other team have. It is depressing to live in a world where being ruthless and greedy gets you further in life than being honest and intelligent, and even more depressing to think that young people will be watching this programme and learning exactly this lesson.

Braided knitting

A few months ago a new sheep, Fernilee, appeared in my flat. (You may remember him from the New Year’s Eve party.) Luckily for him I’m not a very territorial sheep (except for the sofa, which is MINE) but I was a bit put out that he had a lovely hat and scarf and I didn’t. I know that it’s currently summer in Edinburgh, but that isn’t the point.

Seeing as Julia was totally bored after having finished her PhD, I asked if she would make me the scarf to rival all scarves. Not only should it be a warm and functional piece of winter knitwear, but it should embody some sort of mathematical principle so that I can continue inspiring my followers wherever I go. We brainstormed a few ideas: having a stripey scarf where the number of rows of each colour followed the digits of pi (e.g. see here), or having a hidden mathematical image knitted into the scarf (so-called illusion knitting), or having a braid pattern using cabling. The first idea seemed too easy and the second one quite hard, so we decided on a braid.

The braid pattern we chose

Julia had never tried cabling before and wasn’t sure how to design a pattern from scratch, so we decided to find a ready-made pattern for our first braiding attempt. Eventually we decided to go with this one modulo some modifications – changing the border to seed stitch to make it easier, and adjusting the cabling pattern to make the braid alternate (over-under-over-under). It’s a 6-stranded braid with no special mathematical properties (that I can see). In particular, it is definitely not the same as this commutative braid which my officemate Patrick and my old supervisor Andrew are working with.

Being mathematicians (and knot theorists!) definitely helped us to figure out how the cabling pattern worked. In a mathematical braid there are a number of strands running parallel to each other, and every now and then two adjacent strands are allowed to cross. If the strands are labelled 1 to n, then the crossings are denoted by a sequence of numbers, where i means that strand i crosses OVER strand i+1, and -i means strand i crosses UNDER strand i+1. For example, the braid below would be denoted by (1,-2,1,-2):

The braid 1,-2,1,-2

This is in some ways very similar to knitting braids, because in a knitted braid only adjacent strands are allowed to cross. The cable pattern denotes whether the crossing is OVER or UNDER by using F (‘front’) and B (‘back’). The first obvious difference between the maths braids and the knitted braids is that knitted strand-crossings are allowed to occur simultaneously between non-interfering strands. E.g. strands 1 and 2 can cross at the same time as strands 3 and 4 do. Mathematically it makes no difference, but aesthetically it is more pleasing to have simultaneous strand crossings.

The next similarity between maths and knitting is how we ‘add’ braids together. It is simply by putting them side-by-side, the second braid following on from the first. Similarly, the knitting pattern only gives the first 16 rows – the first ‘block’ – and then the braid is continued by placing these blocks next to each other along the scarf.

Braid theory can design paths for these robots so they don't crash into each other

Mathematically, braids are interesting because their addition has a lot in common with numbers. There is a ‘zero’ braid which does nothing when added to another braid – it is the braid with n strands running in parallel. There are also ‘inverse’ braids, which are like negative numbers in the sense that when you add a braid to its inverse you get the zero braid back. (Can you figure out the inverse braid to the (1,-2,1,-2) braid drawn above?) This additive structure makes the collection of braids into a group, and the braid group is of great interest to a lot of people in the world right now! Engineers use them for motion planning in robotics, cryptographers use them to design new codes and computer scientists are using them to design quantum computers.

Braids can also be turned into knots by joining the strands at the end of the braid back to the beginning. I think if I make another braided scarf I shall try to encode the braid for the knot 12a₆₃₁, which is the only knot in my thesis where I couldn’t decide if it was slice or not.

I will end this post by showing you some pictures of how Fernilee reacted to my beautiful new scarf. You can see that he was quite jealous!

Fernilee looks on from behind as I try on my new scarf...

We have a little chat...

He inspects the scarf and admires the mathematical braid

We make friends and play a game of Ninja Sheep!

If you’d like any more information or more photos of the scarf, please get in touch!

Graduation

It’s been just over a week since graduation, and with the official documents in my hands I can finally feel confident at calling myself Dr Haggis.

The 28th June was an auspicious day for a graduation. Not only did the skies produce some amazing sunshine, but it was a mathematically interesting day for at least two different reasons.

The first reason, which actually got a decent amount of news coverage on the BBC and elsewhere, is that June 28th is Tau Day. There is a small but dedicated group of people who believe that pi is the ‘wrong’ number. Not that there is anything wrong with the number itself, 3.14159…, but that we should be using the ratio of the radius to the circumference of a circle, not the ratio of the diameter to the circumference.  So they define tau to be equal to 2 times pi (which is 6.28, hence June 28th!), and have even invented their own symbol:

Tau = 2 pi = 6.2831853...

To modern mathematicians, tau is a more natural constant to use than pi because a circle is defined as the set of points which are a fixed distance (the radius) away from a point. For any of you readers who have studied maths beyond AS level (or Highers), you will know that the standard unit of measure of angles is the radian, not the degree. One radian gives an arc on the circle which has the same length as the radius (see picture). Learning to use radians and converting to degrees can be the bane of a young mathematician’s life! How many radians are in a circle? It’s the circumference divided by the radius, which is 2 pi. So 360 degrees is 2 pi radians. That means we get horrible formulas, like a quarter of a circle being pi/2 radians, instead of what you would expect: pi/4. Using tau instead of pi, we do indeed get a much more intuitive measure for angles.

Still, despite all the arguments in favour of tau, I have to say that I don’t think it will ever catch on. Pi is far too engrained in our history and our culture. We made our choice over 2000 years ago and it’s too late to change things now!

A much better reason, I think, that 28th June is interesting is because it is the perfect day of the perfect month. A perfect number is one which is the sum of its divisors (not including itself). So 6 = 1+2+3, while 28 = 1+2+4+7+14. Perfect numbers, as you might imagine, are pretty rare: the next one after 28 is 496, while the next perfect year is not going to be until 8128. The ancient Greeks were the first people to pay attention to perfect numbers, and it was Euclid who noticed that the perfect numbers had the form 2^p−1(2^p−1), where p is a prime number. He proved that if 2^p-1 is prime then the formula 2^p−1(2^p−1) will always be a perfect number. However, it was not until Euler came along in the 18th century that the converse was shown: that every even perfect number comes from this formula. So every time we find a new perfect number, we find a new prime too!

Me and Nagyi

Graduation was also an exciting day because all of Julia’s family were able to come up and see it. This included her Hungarian grandmother Emily, who we all call Nagyi (which is Hungarian for grandma, pronounced nudge-ee). Although Nagyi could hardly speak any English, she loved the historic atmosphere of the graduations (in the beautiful McEwan Hall) and all the lovely people she met.

I don’t know why this is, but the Hungarian nation produces a surprising number of top-class mathematicians. So it wasn’t very hard to find one at the University of Edinburgh who was happy to chat to Nagyi during the afternoon reception. Our victim was the charming Tibor Antal, who works on modelling biological processes (such as cancer growths) and population dynamics. I hope that he got to tell Nagyi a bit about his research, but I suspect that he spent most of the conversation listening to her life story!

Hyperbolic crochet: add a new stitch for every 4

Another thing which surprised Nagyi was how much everyone appreciated her crochet work here. Before she came, I gave her the double challenge of making a Möbius strip and a hyperbolic disc. Although she made two beautiful models, she had absolutely no idea what they represented or why they were important. And yet, when we took them out at the maths reception, everyone was cooing over them in amazement!

To any other mathematicians who have grandmas: I suggest you get them knitting and crocheting and embroidering any mathematical objects you can think of! They really learn something new from it and have fun in the process. One day I’m still hoping to run an interactive maths knitting exhibition where anyone can come along and try it for themselves.

Hyperbolic disc, McHaggis and a Mobius strip

Thank you once again to everyone who sent messages of congratulation for the graduation, and to all my friends and family for being there on a great day. I wouldn’t have been there without you all!

A week in the life: Wednesday

After another last minute call for help, I found myself on Wednesday helping out with the Maths Extra day that the maths department at Edinburgh were trialling for the first time.

The department usually organises a couple of revision sessions per year for local students studying Higher and Advanced Higher maths. For the past couple of years we’ve used these sessions as an opportunity for a few postgraduates to do short (15 minute) talks on their research. This widens the students’ perspective on what mathematics is, what kind of people do it (for example, women! trendy people! non-bearded people!) and why it is useful.

This year our outreach coordinator, Lois, decided that it made more sense to have the postgraduate talks on a day of their own, together with a set of interesting maths problems that the students could work on in groups. They also got a lecture from our Director of Teaching, Toby Bailey, who worked through a seemingly simple geometry problem to show how an inquisitive nature can reveal many beautiful insights.

I was involved both in giving a talk (about knots in DNA) and in helping to design some of the problems that the students worked on. My favourite problem was one that I plagiarised from the Masterclass Training Day that I was at, where I heard it given by Demi Allen, an undergraduate at St Andrews.

Imagine a very long corridor of rooms, one for each number 1,2,3,4,… To begin with all the lights are turned off.The rule is that on the nth pass down the corridor, you flick the switch of every nth room. So on the first pass, you turn every light on. On the second pass, you turn every second light off. On the third pass, you flick the switch of every 3rd room, etc. The question is which lights are still turned on at the end, and why?

It’s a really neat problem – very easy to get stuck into, and a few different layers to the answer when it comes. I encourage you all to have a think!

My knotty DNA talk seemed to go ok, but the kids were pretty quiet and didn’t ask any questions or interrupt with any remarks. I guess 15 minutes is a bit too short to really explain an aspect of your work AND make it interactive. I’m also getting bored with knots these days and would really like to work on designing some talks in other areas. Longer ones perhaps, with more of an element of discussion around them. Like whether 0.999…=1, or whether Euclidean geometry is ‘wrong’ or just a specific way of thinking. I can feel myself being influenced in these thoughts by Eric Mazur’s lecture (which I’ll discuss in Friday’s post!) – that we don’t just want to impart information but we want students to really think deeply about them too. If anyone has good ideas for discussion points in maths I’d be glad to hear them!

My next speaking opportunity will be on 26th May at Linlithgow Academy, which is hopefully enough time to design a new hour-long masterclass as well as preparing for the science festival and revising for the thesis defence. Time goes so fast these days!