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?

Wizard or mathematician?

“You’re not a mathematician – you’re a wizard!”

This was the verdict delivered yesterday by a group of Dungeons & Dragons fans who had come to ICMS for Doors Open Day, after being treated to some maths busking by me. I also think they went away convinced that I was a geomancer instead of a geometer – I really must work on my enunciation…

Spatulamancy: the art of using a humble spatula to predict the future?

[An interesting aside, geomancy is apparently one of the seven "forbidden arts," along with necromancy, hydromancy, aeromancy, pyromancy, chiromancy (palmistry), and spatulamancy. Ah, I love Wikipedia.]

It’s been a stressful week for me, but culminated in a totally wonderful day of maths communication yesterday. In the morning I gave the first Edinburgh masterclass of the season to a group of 82 enthusiastic 13-year-olds, and some equally enthusiastic student helpers. When I commiserated with them on having to get up early on a Saturday morning, the response was “We’d always get up early for lectures if they were as interesting as this!”. Which is lovely and flattering for me, but really makes me sad that we aren’t doing enough in university to bring our subject alive. Of course not every lecture can be as fun as a masterclass, but there are far too many researchers for whom lecturing is a chore and who never make an effort to bring enthusiasm or interest to their subject.

I digress, but there was an interesting blog post on a related theme by Peter Rowlett this week. He asked whether it was possible to pursue a career in university teaching and lecturing whilst not being a researcher – a question I have full sympathy with as someone in exactly that position. For me the story has a happy ending: after a year and a half of trying to persuade the university that a full time outreach/teaching position was a Good Thing, I have finally got my contract extended to 3 years. It is great to know that the department and university value the things I do, but I would despair of being able to find a similar position were I ever to change universities. While good teaching and public engagement are listed as promotion criteria in many places, in practice they are rarely rewarded when compared with research output.

Another side of the story is that there are many people who do public engagement in their spare time who are not recognised for it. A job title such as mine (Mathematics Engagement Officer) can count for a lot, as my friend and collaborator Madeleine Shepherd has found many times. Although we’ve worked on many projects together, with her often the brains behind the ideas, emails proposing new engagement opportunities are often sent to me and rarely to her.

It was wonderful to see ICMS, where Madeleine works, being open to the public yesterday for Doors Open Day. The building, on South College Street, is a converted church and still has an original stained glass window, among other interesting features.

Doors Open Day at ICMS, featuring Penrose tiles, chaotic pendulum and magnets, Tantrix, and me busking to three D&D fans. Click photo for more ICMS images.

This was the first year it had opened as part of Doors Open Day and we had no idea how many visitors would turn up. In the end I think the count was at 229, most of whom were lured in by the promise of maths puzzles rather than an interest in the building itself. I was only able to attend in the afternoon (due to the masterclass in the morning) and had a huge amount of fun showing people my favourite topological tricks, card tricks and mathematical puzzles. Even those of the public who proclaimed they were bad at maths went away enthused by what they had learnt and wanting to share their new knowledge with friends and family. I hope that we can run such events more frequently instead of waiting for Doors Open Day every year!

This hope is not a forlorn one, as I have big plans brewing… I am currently recruiting undergraduates and postgraduates to be on my new Maths Outreach Team (with unfortunate acronym MOT), and hope to have a team of 10 people trained up and ready to engage by the middle of October. Once they are unleashed on the unsuspecting city of Edinburgh, there will be no end to the school workshops, festival exhibitions, website articles and puzzles, public lectures and impromptu maths busking. At least, that is the plan. If you know of any maths undergrads who would be interested in this, please spread the word!

On that note, it is time for me to head off and hatch more nefarious outreach plans. Please do leave a comment if you were at Doors Open Day, my masterclass, or if you have comments on the difficulties of being rewarded for good outreach and lecturing. Until next time…

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.

Bright Club

When I told friends I was planning to spend the evening watching university academics doing stand-up comedy, the response was a look of confusion and a placatory “That should be…interesting.”

Academics and comedians normally form completely non-intersecting parts of a Venn diagram. After all, what is funny about a gamma ray burst, tree conservation or crayfish? When was the last time a seminar on genetics cracked you up with laughter? To be fair, I do often hear mathematicians making jokes, but normally they are so obscure that only the 3 other people in the room would have a clue why they were funny.

So I really didn’t know what to expect when I went along to Edinburgh’s first Bright Club at the City Cafe on Blair Street. I certainly didn’t expect there to be a long queue at the door because all the tickets were sold out! After a tense wait I was relieved to find myself inside, albeit with standing room only. In a stroke of luck, I spotted my colleague and fellow tweeter Karon McBride who squeezed me in on the seat beside her. She explained that she was quite interested at having a go at the comedy herself and was excited to see how the first session went.

Well, I don’t think that any of you readers are going to be surprised when I say that it was a fantastic evening.  Steve Cross, the founder of Bright Club, came all the way from London to start the proceedings, and we had the enthusiastically foul-mouthed Susan Morrison as our compère for the night. The first academic was none other than fellow mathematics PhD student Hari Srithkantha, which I’m very proud of because it was me who encouraged him to sign up for Bright Club! Hari is already making a name for himself in stand-up, taking part in the Chortle Student Comedy Awards and playing gigs around Edinburgh. However, it was great to listen to him making his research (into gamma ray bursts) the butt of his gags, which I think is something he hadn’t tried before. Even more than that, it was great to find out what he was actually researching! He is probably only the second person (after Matt Parker) to use a graph to make the audience laugh.

Evil American crayfish, likened by Zara Gladman to Madonna

Of the other 7 academics, I don’t think any of them had tried stand-up before, so I was really really impressed with their efforts. Highlights for me were Dan Ridley-Ellis, who talked about the stiffness of wood but managed to avoid all the obvious jokes, Zara Gladman, a zoologist studying crayfish who wrote a song about how they are damaging our ecosystems, and Dan Arnold, who talked about uncertainty and ‘unknown unknowns’. As well as laughing for the whole two hours, I also felt like I learnt a lot about all the science topics on offer, and thought it was an unexpectedly brilliant way of doing public engagement with science.

It is also great to see the nationwide news coverage that Bright Club is getting. The BBC covered the story back in February and the Edinburgh Evening News wrote an article earlier this week. BBC Radio Scotland are running a piece on Friday at 13:15 as part of their Comedy Cafe and there is going to be a BBC Fringe show in Edinburgh on 24th August with all the academics doing the show again. So if you missed it then don’t worry, there will be a chance to catch up – but only if you’re quick! Tickets for the BBC show are only available until 8th August (despite what it says on the website) so make sure you sign up pronto!

Bright Club is going to be a monthly event in Edinburgh, so I look forward to seeing Karon and other academics getting on stage and making people laugh with their research. It’s going to take a lot more persuading to get me to think about having a go though!

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: Friday

Gosh, I’ve not done very well in keeping up with this series of blog posts, have I? For the past week I’ve been caught up in the Edinburgh International Science Festival, helping to chair some talks and run an exhibition at the National Museum of Scotland about game theory. More on those in another post perhaps.

So, Friday. Although it’s been over two weeks since this particular Friday, I remember it very well. It was on this day that we were lucky enough to have a visit by the distinguished professor Eric Mazur.

Mazur & Me

Although Mazur is distinguished in his field of physics (lasers, semi-conductors, optical fibres), it wasn’t a physics lecture that everyone turned out to hear. Surprisingly (even to himself) he has become most well known for his radical teaching method, known as peer instruction. The talk that he gives about this is really fantastic, and I recommend that everyone watches it on YouTube.

It is ironic that Mazur should be touring universities around the world, giving lectures about peer instruction, when a fundamental tenet of the theory is that we shouldn’t be lecturing to students! The idea is that people don’t take in information when they are forced to sit and listen to something; they have to be doing and discussing the subject matter in order to really engage with it. This became especially clear when Harvard University physics students were given a simple exam which tested their basic understanding of Newtonian physics.

Big truck vs little car

For example, suppose that a heavy truck and a small car crash into each other. At the moment of impact, is the force of the truck on the car (a) larger than, (b) smaller than or (c) equal to the force exerted by the car on the truck? Think about this for a moment…….. Your intuition is probably telling you that the truck exerts a larger force on the car than the car on the truck. Yet anyone who can remember their high-school mechanics should know Newton’s 3rd law: that forces are always equal and opposite.

What was interesting was that Harvard physics students got this question wrong almost as often as the general public did. It’s an extremely strong indication that students are only superficially learning information, memorising things in order to pass an exam but not really internalising the concepts. We see this all the time with our maths undergraduates too. They can compute all manner of difficult integrals and solve complex matrix equations, but in the end very few know what the answers mean or why they are important. What actually is the determinant of a matrix, or what does it mean to have an infinite decimal expansion?

But if lecturing doesn’t work as a means of education, then what else can we do? Eric Mazur’s answer is peer instruction, which works something like this. The students read the material in a textbook before the lecture, submit a list of things they don’t understand to the lecturer, then during class the students work through questions designed to address and correct their misconceptions. Questions are often presented in multiple-choice format and students have the chance to vote on an initial answer before discussing with their peers and then voting again on a new answer. From research into this method, it seems that students teach each other a lot more effectively than a lecturer can. They understand each other’s problems and can more easily get to the heart of the explanation. And nobody can just sit and sleep through the lecture, because there is constant discussion of the material in the class.

At Edinburgh, we would love to try and implement this method with our first year undergraduates in September. The main difficulty in starting out is getting those multiple choice questions which can really change people’s opinions about a subject. What are the common misconceptions in maths? When does our natural intuition override the definitions we are given in lectures, like in the physics example before?

The only university which has really implemented teaching like this in mathematics is Cornell University, and you can take a look online at their list of ‘Good Questions‘. Here’s one to get you talking:

Was there a time in your life when you were exactly pi feet tall?

A good question, to me, is one where you have an instinctive immediate answer, but then when you think more carefully you really get to very deep questions about the subject.

Can a person ever be exactly pi feet tall?

For example, someone might immediately think that the answer to the above question is “no”, because pi is an infinite non-repeating decimal and nothing can be ‘exactly’ pi feet long. But then they might think, is it really because pi is irrational that the answer is “no”? Was there ever a time when they were exactly 3 feet tall? Or they might think, there was a time when they were less than pi feet tall and now they are bigger, so surely there must have been a moment (however brief) when they were exactly pi feet tall. Discussions of this will get to the heart of the real number system and questions of approximation, which are essential for anyone studying analysis to master .

I would be really interested to hear from anyone with opinions about this. Do you remember which concepts in maths you struggled with the most? When do you think you learnt your specialist subject: through listening to lectures, or at home with a textbook, or chatting to friends? Do you believe that peer instruction can work or do you think the system is fine as it is?

Send me your comments!

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!

A week in the life: Tuesday

Continuing with my hectic life in the last week of March…

Tuesday was a day for me to prove my worth as the next David Attenborough or Brian Cox.

The College of Science & Engineering had been successful in getting some money from the research council EPSRC to produce a series of 5-minute videos highlighting the work that was being done by scientists at Edinburgh. Each department was supposed to nominate one or two of their researchers to speak for the camera, highlighting the applications and benefits of their work.

A supercomputer

Now, before you start thinking that I am a fame-hungry media-whore of a sheep, let me stress that I (and my maths colleagues) did what we could to find ourselves a real bona fide applied mathematician to do this. At the beginning of the year we approached Jacek Gondzio, who works in an area of maths called optimization. Jacek made it into the Guardian in 2008 for using Britain’s fastest supercomputer to solve problems relating to financial mathematics and risk modelling. Clearly his work is more important now, in our financial crisis, than it has ever been, and it would have been great for him to have the chance to get on camera and explain it to the wider community. Sadly, despite our urging, Jacek was just too shy to take up the opportunity.

This reminds me a little of a joke: How do you spot an extroverted mathematician? Answer: They look at your shoes when they’re talking to you.

A black hole of filmable mathematicians?

Of course not all mathematicians are shy and socially inept, but it was certainly going to be a challenge to find someone who was extroverted, articulate, passionate, and working on something useful. And in Edinburgh. And with an EPSRC grant. We ended up, with 2 weeks to go before the filming date, with a shortlist of 4 people, of whom one was out of the country, one was too quiet and another was too busy. We asked Joan Simon, a charismatic Catalonian who works on the theory of black holes, but after a few days’ consideration he decided that he was also too busy.

At this point, things were dire enough that I suggested to Julia that maybe we should do it. It was a bit of a long shot, seeing as we were no longer EPSRC funded and that our research had barely a sniff of real-life about it, but it still seemed better for us to do it than for the opportunity to be wasted. And we had loads of great props that really deserved to make it onto the big screen.

Tuesday morning came and we didn’t really know what was happening. We’d submitted a script to the producers and booked a room, but nobody had confirmed that we were doing it. Julia had some quite impressive bags under her eyes from staying up late to rehearse the script and I had specially combed my quiff, just in case. Finally, an email! “Be there at 11:30.”

The film ‘crew’ were two PhD film students, one of whom (Alastair Cole) had his own documentary company specialising in linguistic anthropology. Wow. I think they were a bit apprehensive about filming mathematicians, but after they caught sight of the sheep and knitted surfaces, the apprehension turned to curiosity and amusement. Sadly I wasn’t allowed to be in the film (and my quiff was so beautiful too!) but the video did not end entirely sheepless…

Knitted torus (aka doughnut) and sheepy coffee cup

We (well, Julia) started off explaining what a topologist was: someone who looks at those properties of objects that don’t change after stretching and wiggling. This gave us the opportunity to stick in the classic joke that a topologist is someone who cannot tell the difference between a doughnut and a coffee cup, and to exhibit said objects. We then explained that what concerned knot theorists was not so much the objects themselves but how they sat in space (the embedding). All mathematical knots are intrinsically circles; it’s the way they sit inside 3D space which is interesting. And such work is going to have applications in looking at objects in our universe. For example, how do black holes sit in our 4D universe?

We then had to find a way to explain slice knots. That is, knots which are slices of spheres sitting in the 4th dimension. What even IS the 4th dimension? How can we visualise it? With more knitting, obviously. Every knot can be drawn as the edge of a surface, called a Seifert surface. It’s pretty easy to picture a surface for the simplest untangled knot: it’s just a disc, like a frisbee. Picturing a surface for the overhand knot, or trefoil, is already much harder. One surface which works is the 3-twisted Möbius strip, although this is not strictly allowed because it has only one side. You could also picture 2 discs, drawn one above the other, with three twisted strips joining them. (Pictures below courtesy of Seifertview.)

A trefoil knot bounding a 3-twisted Möbius strip

A trefoil knot bounding an orientable (2-sided) surface

Or, if you had the time, you could crochet them!

Crocheted one-sided trefoil surface

Crocheted two-sided trefoil surface

Each of these surfaces is very different from the frisbee because there are holes in the surface. Sometimes there is a way of pushing the surfaces into another dimension so that the holes go away – such knots are exactly the slice knots. We demonstrated this rather abstract concept using a 3-dimensional analogy…and a sheep! You’ll just have to wait for the final video to see how we did it.

Hopefully the film will be edited and put online (on the University’s Youtube channel) within the next month. When it is, you lucky readers will be the first to know!

I’m looking forward to getting other mathematicians on board in the next round of filming. Hopefully they will be less daunted by the prospect when they see somebody else doing it. And hopefully seeing the finished edited film will help me in starting to do some filming of my own.

Look out for future blog posts about mathematical knitting – it’s all the rage these days! We’ll be using knitted torii to play noughts & crosses at the Edinburgh International Science Festival and I’m hoping to make some (slightly better) Seifert surfaces for different knots. If you have ideas for other projects, let me know!