Most random masterclass question ever

Would your DNA be long enough to weave into a tea towel?

Yesterday I was in Stirling giving my usual knots masterclass when I got asked the most unusual question ever! I was explaining how DNA is like a very very long piece of string sitting inside each of our cells. (It’s actually about 2-3m in each cell!) One pupil then put their hand up and said “If you took all the DNA from all the cells in my body, would it be long enough…to weave into a tea towel?” I was so taken aback by this that I just laughed, but actually the answer is probably yes! Depending on the size of tea towel that he wanted of course. And it would be so thin that it wouldn’t be very good at drying dishes, but that is beside the point.

It is always a pleasure to be surprised by children. And I was especially impressed at all the questions I was asked in Stirling. It helped that the masterclass was much smaller than usual – 15 pupils compared to about 70 in Edinburgh and 40-50 in Glasgow. It meant that I was able to talk to everyone while they were doing exercises, and it was much less intimidating for people to stop me and ask questions during the presentations. But even given those things, I was genuinely taken aback by the insightful questions I was asked and the interest that they showed in what I had to say.

My other favourite moment was when another child pondered “I could take a one-dimensional piece of string and weave it into a 2-dimensional object…”. (I assume this was inspired by the previous question about tea towels.) Now, I had not mentioned the word ‘dimension’ once in my whole masterclass. Most people don’t understand the concept of dimension. Even the undergraduates I teach would have trouble understanding why a knot is inherently one dimensional and not three dimensional. And here is a 12-year old child explaining to me how you could use one dimension to fill out 2-dimensional space!

In case you still don’t understand why this bowled me over, you should go and Google “space filling curve”. (Wikipedia is not a bad reference, but is a little technical.) In the mid-19th century, mathematicians had the idea that a curve could be drawn inside a square so that it went through every single point of the square. This is counter-intuitive, as it seems like the square is ‘bigger’ than the curve, so how could the curve fill it all out? Cantor had showed in 1878 that the infinity which is the number of points in a line segment is the same as the infinity which is the number of points in a square, but it was not until 1890 that Peano came up with this geometrical argument that demonstrated it.

How to construct a space-filling curve iteratively. After infinitely many iterations, it will fill out the whole square.

It is still very counterintuitive to mathematicians that this curve is continuous (i.e. can be drawn without taking your pen off the paper) but is nowhere differentiable (i.e. every point is a ‘corner’, so the curve is always changing direction) and is everywhere self-intersecting (every point on the curve touches another point on the curve). Maths is full of these great examples that challenge our assumptions and intuitions and I hope that I can teach this to my undergraduates later in the semester.

So although the masterclass pupil probably didn’t have infinities and deep thoughts in mind when he made the comment about weaving, it’s exactly questions like this which got mathematicians discovering such things over 100 years ago. I hope that his teachers continue to encourage this wonderful imagination and willingness to ask questions, however silly they may seem at first.

MathsJam 2011

This post is a VERY long overdue one… I had meant to write a post after the (very first) MathsJam conference in November 2010, then after the first few Edinburgh MathsJams… We’ve now had the second national MathsJam conference and 5 Edinburgh ‘Jams so it’s about time I told you readers what it’s all about!

Balloons at MathsJam 2010

In a nutshell, MathsJam is a place for people to meet to share mathematical puzzles, games, toys, ideas, stories and tricks. It was originally the brainchild of Colin Wright, who organised the first ever MathsJam conference last year, bringing together geeky enthusiasts from all over the country for a weekend of mathematical fun. We had Rubiks cubes of all shapes and sizes, mathematically folded balloons, mirrors, Klein bottles, magic tricks, soma cubes, post-it note dodecahedra, vortex cannons… And that was just the list of physical toys! I also learnt facts like:

• Round pegs fit into square holes better than square pegs fit into round holes…until you get to 9 dimensions!
• Almost every integer contains a 3.
• That given 5 numbers, you can always find 3 of them which add up to a multiple of 3. (But what is the generalisation?)
• That there is only one number whose spelling is in alphabetical order. (Can you find it?)
• That a blindfolded person given an even number of coins, placed on a table so that half are facing heads up and half are facing tails up, can separate them into two piles so that the number of heads in each pile is the same.
• That you can work out the distance to the moon using only a pendulum.

The weekend was such a success that people started asking “Can’t we have a MathsJam every month?”. Pretty soon there were ‘Jams in Manchester, Nottingham and London with Edinburgh, Glasgow, Reading, Liverpool, Newcastle, Dorset, Leeds, Bath, Dublin and Belfast following on their tails.

Attempting topology at the August Edinburgh MathsJam

Everyone meets on the second to last Tuesday of the month and we have a shared Twitter account, @MathsJam, so that everyone can see the puzzles being worked on around the country. The Edinburgh ‘Jam was set up by myself and Ewan Leeming, and we meet at Spoon Café Bistro on Nicholson Street. Further details are on the MathsJam website, together with an email address and Facebook page, and also contact details for all the other ‘Jams around the UK (and indeed, the world!).

This weekend I travelled down to somewhere near Crewe for the second annual MathsJam conference, together with my buddies Albert, Julia and Michael. I was very excited about all the toys and games I’d get to play with, but at the same time incredulous that the weekend could possibly be better than the first MathsJam weekend. Well, I shouldn’t have had any such thoughts.

Albert wearing his ring-on-a-chain

One thing I loved about this year’s conference was the chance to purchase goody bags with exciting toys to take home and show friends. Last year I shot some videos and got photos, but nothing compares to being able to go home and show your friends in person the amazing things you’ve seen. My favourite was the ring-on-a-chain trick (pictured left) where a ring is dropped from a chain with unexpected consequences. Next favourites the falling rings and James Grime’s amazing non-transitive dice.Maths and science is much more cool than sleight-of-hand magic.

Here are some pencil and paper questions you might like to get your teeth stuck into (metaphorically speaking):

• A consecutive sum is a sum of consecutive digits. Are there any numbers which are not consecutive sums? How many ways can a number be written as a consecutive sum?
• Why is 100/81 equal to 1.2345678…?
• How can you cut any shape out of a piece of paper using only one cut?
• Does a running sand timer weigh more, less or the same as a finished sand timer?
• How do you make 2 paperclips link together using a strip of paper?
• Given that we can make a regular pentagon by tying a knot into a strip of paper, is it possible to make a dodecahedron by folding 12 knots into a piece of paper and then folding it up?
• How is it possible to randomly play two games, each of which would individually lose you money, and make an overall gain? (This is called Parrondo’s Paradox.)
• Split the numbers 1,..,16 into two sets X and Y so that the sum of the elements in X equals the sum of the elements in Y; the sum of the squares of X equals the sum of the squares of Y; the sum of the cubes of X equals the sum of the cubes of Y. (I am currently working on a generalisation!)

Plus I learnt  that a 9999-sided polygon is called a nonanonacontanonactanonaliagon. (This seems to be the most popular thing I have ever posted on Twitter.) I encountered Pat Ashforth, one of the founders of Woolly Thoughts, who showed me her dragon-curve blankets and crocheted hexaflexagons. I also saw a magic square that worked upside down and some Platonic solid maps of the world.

Dragon curves and other mathematical knitting by Pat Ashforth

A magic square cushion which works both ways up

Julia found herself on the panel for the Math/Maths podcast, which you can listen to here,with contributions also from Matt Parker, James Grime and Katie Steckles. The laughter on the podcast is a really good reflection of the fun that everyone had at the MathsJam, and once again I have to extend a huge thank you to Colin and all the other people who helped to organise the event this year. There’s no other conference in the world which is this enjoyable and it is wonderful to see so many people enjoying the fun and beauty of mathematics.

If you’ve never been to a MathsJam, I hope this article persuades you to go along to the next one on 22nd November! They are all over the country now so there’s bound to be one nearby. And if there isn’t, start one up yourself! All you need is a pub and a couple of people willing to come sit with you on a Tuesday evening. I look forward to seeing more people MathsJamming in Edinburgh in a week’s time!

Clicking infinity

On Wednesday I was asked to give a talk to a small bunch of S6 (Yr 13 in in England) pupils who were visiting Edinburgh from Fife. It wasn’t any particular special occasion – the enterprising teacher just wanted his students to get out and learn some exciting and different mathematics. It was the perfect opportunity for me to try out a new piece of technology that I’d heard my boss raving about: clickers.

A clicker

A clicker is like one of those things they have in Who Wants to be a Millionaire where the audience votes for what they think is the right answer. It is an absolutely wonderful teaching aid and we are very lucky to have them at the University of Edinburgh. It means that students can tell the teacher their thoughts without letting anyone else know what they are thinking, so they needn’t worry about the embarrassment of having the wrong answer.

The clickers we use are exactly those pictured on the left. There are 6 buttons which can be used for multiple choice questions, and also a True/False option. The software that comes with the clickers is capable of storing a huge amount of data about your sessions, which really comes into its own when you are monitoring a specific class over many weeks rather than just having impromptu sessions. You can see whether students are improving, how often they change their minds about questions, and even (if you have a strict seating plan) how ideas are spread around the classroom.

I decided to make my class about infinity, using the story of Hilbert’s Hotel to hold the plot together. My first question, just to get people used to the clickers, was a simple true/false question: “Infinity exists only in our imaginations”. There was a fairly strong preference for ‘false’ from the class, which was interesting for me because I would usually vote the other way myself. The students gave examples of ‘real’ infinities that I would argue are purely abstract mathematics, such as the infinity of numbers or infinities in fractals. It’s actually quite nice to meet people who believe that abstract thoughts are as real as anything else in life.

Take 2 balls out, put 1 back, repeat. How many are left in the end?

The surprises didn’t stop there, and I really believe it was the clickers which made the session work. One of my favourite infinity questions is the ‘balls in a barrel‘ paradox which I learnt from the ever-wonderful Colin Wright. If balls numbered 1,2,3…etc are in a barrel, and at each time step two are taken out and one replaced, then after infinitely many time steps how many balls are left in the barrel? (a) None (b) All of them (c) One (d) Half of them, or (e) Not enough information to decide? As I’d hoped, people were very split on this question, with a small majority going for either (b) or (d). A lot of the students were very shy and I don’t think they would have volunteered an opinion without being able to do it anonymously. But once they saw that nobody else really knew the answer either, they were more inclined to speak up in favour of the option they had voted for, and we really got a great discussion going. (If you don’t know the answer yourself, have a good think about it before reading Colin’s article!)

My favourite clicker question was at the end, where I was basically proving uncountability. (I got the idea for how to incorporate this into Hilbert’s Hotel from an xkcd chat forum!) We’d got to the point where I’d done the diagonal argument and asserted the existence of an element which was not in the infinite list we had assumed contained every element. I asked them if they agreed with this. Usually when I teach this I just assume that the argument is crystal clear, and students usually nod and smile. This time, using the clickers, I found that the class were exactly split each way! Half agreed that the new element was definitely not in our list because of the way it was constructed, while half asserted that it must be in the list, because that’s what we assumed at the beginning. Once again we got to have a great discussion, examining our implicit assumptions and coming to the mind-bending conclusion that there are different sizes of infinity!

Get people to vote using fingers, then hold them against their chest so nobody but you can see

I really hope that more schools and universities will start using this method of teaching. You don’t even need the fancy technology – some voting cards or fingers against chests are adequate for the purpose. It comes into its own with the quieter members of the class, giving them a voice they otherwise wouldn’t have had, and pushing weaker students to have opinions about things they’d otherwise not bother thinking about. There’s certainly a skill in asking the right questions and in not being scared to ask things which seem obvious to you. For example, my boss Toby asked his class  “2 ≤ 3, true or false?”. Stupid question, right? But half the class disagreed, asserting it was wrong because two is less than three, not less than or equal to.  Misconceptions occur at the deepest levels and we must work hard to root them out!

I’d love to hear other people’s stories of using clickers or other similar teaching methods. What have been your most surprising results?

New blogs

Topology on a blackboard, courtesy of Ryan Budney

Haggis the Sheep is back on the blog! And not just this blog, but two others that I’ve started up. The first is a photo-blog about mathematicians’ blackboards: What’s on my blackboard? Every week I want to upload a photo of a blackboard with some interesting or beautiful (or both!) maths on it, along with a short description of the mathematics. I think that there really is something wonderful about seeing the random scribblings of a great mathematician, or seeing the beautiful abstract pictures that we draw. Spread the word and get your local mathematicians to send me photos! (And it can be whiteboards too – I’m not discriminating!)

The other blog is to document a project that my friend Madeleine Shepherd and I have just got funding for. It’s called The Mathematician’s Shirts and is being funded by ASCUS, the Art Science Collaborative in Edinburgh. Madeleine and I beat off a lot of competition to secure the funding and we’re really excited about getting started on the project.

A humble shirt, but what will it become?

The idea here is that we are going to make a series of soft sculptures out of shirts to represent different mathematical concepts. For example, we could pass a shirt’s sleeve through itself to make a Klein Bottle, or we could sew successively smaller sleeves onto each other to make a fractal object. It was Madeleine’s idea to use the shirt, since it is an iconic piece of clothing, representing the formal and largely male world of mathematics. Perhaps some of the shirts will be donated by mathematicians themselves!

Here’s the timeline for the project. Over the next few weeks we’ll get together with local mathematicians to brainstorm ideas and make a concrete plan for between 5 and 7 sculptures. Then in September and October it’ll be time for the practical work to begin, actually sewing and making the sculptures in Madeleine’s studio. Finally, in November there’ll be an exhibition in a ‘non-standard’ location. That is, not a maths department or a science museum or an art gallery. We thought maybe we could have our exhibition in a shop window to entice passing shoppers.

If you have ideas on either of my two new projects, I’d be very glad to hear from you!

Doctor Haggis!

Dr Haggis, the knot surgeon. Not to be confused with real doctors who do surgery on people.

Well, one step closer to Dr Haggis at least! As you may have surmised from the comments and Twitter feed, I did indeed pass the viva yesterday, and have just some minor corrections to complete before I can be officially awarded my title. I promise to use it responsibly!

As expected, the defence was quite enjoyable, and it felt more like an extended seminar than an exam. We got off to a late start because the ‘Non-Examining Chair of the Examination Board’ (thanks Laura!) didn’t show up. Since the internal examiner Mark had never examined a thesis before, we needed a third person to keep an eye on him; eventually a substitute was found, and it was Chris Smyth, who is actually my second supervisor. I really hope there isn’t some rule against this which makes the viva null and void!

So after that brief faff I began proceedings by giving a little presentation summarising the aims, importance and main results of the thesis. Although intended to be only 15-20 minutes long, it was easily 45 minutes with all the interruptions and questions by Brendan and Mark!

I thought it would be a bit pointless summarising results which they already knew, having read carefully through the thesis, but of all the chapters I think it was my main results chapter which they hadn’t read in detail. Disappointingly, they didn’t spot the mistake in one of my big proofs, which I had frantically spent the last week trying (and eventually succeeding) to correct! It does make me wonder how many incorrect results actually manage to pass through vivas and referees, and how many never get noticed. Is it any better in the other science disciplines?

Talking about my own results was the easiest part of the viva. The kinds of questions I failed at answering were the ‘elementary’ ones. Results that are written in so many textbooks that you take them for granted without making the effort to understand the proof. And there were even a couple of questions that none of us, even the examiners, could answer! Hopefully I will sort these out in the coming week.

Brendan Owens, Julia and Mark Grant. I am wearing my lucky Seifert surfaces.

At this point I’d like to say a big thank you to Mark and Brendan for being such great examiners and for helping me to feel relaxed about the whole thing. They caught lots of my mistakes but also gave me much-needed encouragement that my results were important and interesting.

The viva was over 4 hours in the end (fairly long for a maths defence) and poor Andrew (my supervisor) was pacing the corridors ‘like an expectant father’ (in his own words). There were many sighs of relief when the examiners finally delivered the verdict, and much drinking of alcohol afterwards! Thank you also to all those who emailed, texted, tweeted or otherwise conveyed messages of congratulations, especially little sister Suzanne who seemed very emotional about it!

I now have 8 days to complete all corrections, get the thesis printed, bound and signed and handed in, so that I can graduate at the ceremony in June. Of course, if I miss the deadline it’s not the end of the world, as I can graduate in November instead, but all of Julia’s family are coming up to Edinburgh in the hopes of seeing some be-robed students, so I had better make the effort. It’ll be a tight deadline, but since the corrections are all pretty minor I think I can do it if I work hard.

And then I shall be free! I pledge now to

• Do more exercise
• Do more blogging
• Do more mathematical knitting
• Do more exploring of interesting places
• Do more exciting public engagement things
• Do more keeping up with poor neglected friends
• Do more cooking of healthy things
• Do more enjoying of the beautiful Edinburgh summer weather

But for now, I shall simply be doing more sleeping!  G’night all!

P.S. Congratulations are also due to my mathematical brother Mark Powell, who passed his viva on Monday, beating me by 2 days. Grr.

Judgement day!

Well, this is it. Half an hour to go until we begin the 3 hours that will judge the past 4.5 years of my life. Yes folks, it is my thesis defence!

It is pretty nerve-wracking for anyone to have their work closely scrutinised by a group of experts, but I think it is especially so for a mathematician. The standards of rigour and clarity are exceptionally high, and it is easy to work on a problem by yourself for a long time and not see subtle errors creeping in. Myself, I’ve not talked about my thesis since last summer, so I am expecting lots of corrections from my examiners. And to have to explain things in detail that I’ve never had to consider before.

Despite the nerves, I’m pretty excited too. As mentioned in a previous post, this is the one time in my life when I get to talk about my work to people who have spent a month reading it and trying to understand it. Perhaps we will even shed new light on some of the problems I failed to solve.

And on top of the fear and excitement is a sort of melancholy.  It’s the end of an era today; the end of being a student, the end of being a research mathematician (at least for the time being) and the end of thinking about all the problems in my thesis. I may go back and think about some of them again, but realistically this is unlikely to happen unless I am surrounded by other mathematicians interested in the same ideas.

Many thanks to all the people who have wished me (and Julia) luck via email, Twitter and Facebook. Your support means a lot, and I will report back with the result soon enough. Fingers crossed!

A week in the life: Thursday

The thesis

Thursday was probably the most exciting day of my week, because I handed in my thesis! All 131 pages, lovingly printed and bound and signed. It felt like something of an anticlimax actually, since I just handed in the document and that was it. No ceremony, no party, nobody bowing in deepest admiration. No ritual to welcome me to the club of people who have overcome their struggles and finally submitted.

My supervisor Andrew Ranicki was kind enough to walk over to the building with me and see the happy moment. I suspect that he was even more happy about it than I was, and didn’t want to risk anything going wrong in between the printing and submitting. He had a double whammy that week actually, since my mathematical brother Mark submitted his thesis on Wednesday. (Which was REALLY sneaky: I knew he had originally planned to hand in on Friday, but after learning that I would hand in on Thursday he decided to beat me to it. Grr.)

Now what? That’s it, right? All written up and handed in. Time to party until graduation, no?  This is at least what many people seem to think, so let me me explain the PhD process for anybody who doesn’t know it.

Mark Grant, my internal examiner

Brendan Owens, my external examiner

After submission, the university sends your thesis off to two people who have been chosen to examine you. The first person is a leading expert in your field and generally comes from another UK university. In my case it is Brendan Owens from the University of Glasgow, and he is called the external examiner. I’m really chuffed to have Brendan as my examiner because he’s a lovely bloke as well as being a top-class researcher in knot concordance. In fact, he’s already emailed to point out one small oversight in my thesis, so I know he’ll do a thorough job! The second examiner, called the internal examiner, is chosen from within your own department and should be as close to your field of study as possible. In this role I have Mark Grant, who is a young topologist doing work on motion planning in robotics. He’s not an expert in knot theory but should be able to offer valuable suggestions on the work in the thesis.

And because Mark is young enough not to have ever examined a thesis before, there will be someone at the exam to keep an eye on him! I don’t know what the official name for such a person is, but I’ll have José Miguel Figueroa-O’Farrill doing that job. He isn’t allowed to ask me any questions himself, but is just there to keep an eye on Mark and make sure he’s asking appropriate things. I don’t think he even has to read the thesis if he doesn’t want to.

A PhD examination is called a defence or viva (short for viva voce, which means “live voice” in Latin) and will probably last between 1.5 and 4 hours. (I do know people in other subjects who’ve had 6-hour defences though!) The examiners go through the thesis, asking questions about parts they didn’t understand and making sure that the candidate knows their stuff. They need to be satisfied that the student really did all the work themselves, that they understand all the ideas behind the research and have made a genuine contribution to human knowledge.

My defence is going to be on the afternoon of 11th May, which really doesn’t seem long away given all the stuff I need to learn by then! You’re probably thinking “You wrote it, you surely understand it!”, which is a good point, except that I wrote a lot of those ideas down more than a year ago. And I currently feel less than confident about a lot of the background material to the thesis. There are some very difficult ideas behind slice knots which I regret not having spent more time learning back in the first couple of years of my PhD. I will do my best to revise before the big day, but I am absolutely not taking the defence for granted. Many people tell me it’s just a formality, and my supervisor wouldn’t have let me submit if it wasn’t good enough, but I disagree and think it’s all still to play for.

Having said that, I am also looking forward to the defence. In one sense it’s going to be the most difficult and intense exam ever, but in another sense it’s an opportunity to discuss my work in depth with experts in my field. This is an opportunity I’ll maybe never have again. Nobody is ever going to be forced to read my research after this! I’ve not had to explain my research to anybody for a really long time – not since last summer when I visited Chuck Livingston (my unofficial supervisor over in Indiana), so I’m both looking forward to seeing what people think of it and scared of the criticism it’ll come in for!

So, many thanks to all the people who congratulated me on handing in, but I will be saving all the celebrations for the moment when Brendan and Mark tell me that my thesis is good enough to earn me the title of Doctor!

Addendum: A thesis defence in the UK is not nearly so intimidating as one in the Netherlands, where it is a public event with black-tie dress and about 7 examiners. But some students take advantage of the occasion to do something a little out of the ordinary, such as dancing their PhDs! Take a look at this wonderful video of a knot theory student putting their PhD to music (from about the 1 minute mark). Amusingly, the two helpers at the ceremony are called paranimfs.

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!

A new academic year

September has come around again, and with it a new influx of students into Edinburgh.  I like the sudden hustle and bustle of the university, even if all the lecturing staff seem completely stressed out at the moment!  Last week we had a couple of events to welcome the new postgraduates into the maths department.  On Tuesday was the departmental welcome party, where I got to meet the new staff and students for the first time.  I started chatting to an ex-number-theorist-turned-probability-student, and suddenly two hours had passed and the whole blackboard was covered in knots and other crazy symbols.  I love it when that happens!  I learnt a lot of stuff and also felt completely motivated about my own research again.

Watching someone enthusiastically talking about their work, clearly in complete thrall of the beauty of the subject, is something I will never get bored of.  And it’s something I hope to capture in the lecturers here at Edinburgh, by interviewing them and filming them talking about their research.  The ‘Outreach’ pages I wrote for the maths website are finally online, and I have a section called ‘Spotlight on research‘ where I will be posting these interviews.  So far I’ve only done one, and I haven’t yet got a video camera to film the interview, so I hope the written article will do the subject justice.

Speaking of videos, I thought that before I go ahead and try doing professional filming, I should at least try making and editing a little video at home.  My idea for a short clip was to document all the many sheepy items that we have in this flat, because there now seems to be quite a lot!  I persuaded Julia to help me out, and this is the result:

The most difficult thing was not in making the video but in editing it afterwards.  This version was put together by Windows Live Movie Maker, mainly because I haven’t yet found some decent open-source software to use.  It would have been nice to edit the audio separately from the video, which WLMM isn’t able to do.  Suggestions welcome!

The final thing I want to mention from this week is my meeting with a young music student.  Harry came to see me on Friday morning, having just started his PhD and wanting ideas for a composition.  Topology is not a stranger to the music scene; composers like Bach, for example, made music containing a Möbius-strip-like structure.  Harry’s idea was to get the maths behind a knot into his music.

“How on earth do you encode the structure of a knot into a piece of music?”, was my immediate response.  Thinking about it for a bit, I thought that maybe you could encode some symmetries of a knot (such as mirror-symmetry or orientation-invariance) into the music.  Then I thought that maybe it would be easy to make music from a braid, since you’d just need a few interweaving melodies.  But when Harry came along, one of the first things he said was “If you just give me a matrix then I can turn that into a composition.”  Well, that was all the encouragement I needed! My thesis is very much built on the idea of making a matrix from a knot and using that to tell me about the knot’s 4-dimensional properties.

20 minutes later and Harry’s notebook was filled with my attempts at trying to explain Seifert surfaces, Seifert matrices and slice knots.  I mentioned to him how I was very impressed at his speed of understanding; he seemed to pick up concepts that other people (with more of a maths background!) would struggle with for hours.  Slightly bemused by my comment, Harry thought about it for a few seconds, and then said that it wasn’t so strange after all, because mathematicians and musicians think in the same way.  Both subjects are highly creative and abstract, involve writing down symbols and (at least in mine and Harry’s case) thinking of concepts in a very visual way.  When he composes music, he ‘sees’ it in the same way I can ‘see’ my 4-dimensional knots.

I hope that together Harry and I can do more to put music and maths together and to popularise both subjects.  I will let you know if more developments happen on the subject!

A new friend

Hi everyone, I hope the summer is going well for all of you.  It’s been a little boring of late for me, what with Julia going away to America for a week and then being boring and writing her thesis when she got back.  How I have been longing to climb some hills or go exploring somewhere, but all I can do is sit here on the sofa philosophising and daydreaming.  And Tweeting, of course.

So it was with great excitement that I watched Julia open her birthday presents last week, and saw this little creature emerge from the packaging:

She was a gift from my old friend and former housemate Graeme, and her label says that she is called Kissing Disease.  “What on earth…?” I hear you begin to ask.

She comes from a company called Giant Microbes who make all of your favourite diseases and complaints in fluffy format.  For example, you can also get The Common Cold, Bad Breath and The Plague among hundreds of others!  The Kissing Disease also goes by the names of ‘mononucleosis’ and ‘glandular fever’, and apparently more than 95% of the population has been infected by it.  Most people catch the virus when they are very young, and will barely notice that they have symptoms, whilst others (Julia included) catch it as young adults and suffer from a terribly painful sore throat and a general malaise which can last for up to a year afterwards.  The virus is a pretty sneaky one and is most contagious in the 4-7 weeks before any symptoms start showing, so infected people are going around kissing each other and passing it on unknowingly!

As I said, I was very excited to see this creature because it meant that I had a new friend to share my sofa with!  So far I’m having trouble thinking of a good name for her, and am calling her Betty until I can come up with anything better.  Suggestions from you guys are more than welcome!

Seeing as Julia has been neglecting me so much, I feel no guilt in revealing her latest secret, which is something that she’d rather keep quiet from all her mathematical friends.  It is that, in her free time, she has been indulging in some tv-watching and, in particular, watching episodes of Britain’s Next Top Model.  Who would have thought it from someone so fashion-ignorant?  I feel particuarly indignant, since it is quite obvious who Britain’s next top model is.  Me.  ME!  Did you ever see anyone as photogenic as me?  No.  I rest my case.

After I pointed this out to Julia, she agreed to play the role of photographer (in the absence of the talents of Graeme) and do a little photoshoot with me and my new friend Betty.  Here are some of the results:

And here’s us relaxing in my favourite place: the sofa.

What do you think? Do we have what it takes to be models?  (You can click on the pictures to get bigger versions and see me in all my close-up glory.)

This weekend I can look forward to some more action in the flat, since we are being visited by the science communicator and postgraduate singularity theorist Joel Haddley.  He’s coming up to see some festival shows at the Edinburgh Fringe, including the wonderfully funny Matt Parker doing a show called “Your days are numbered: the mathematics of death”.  If you’d like to join us, we are planning to see the 11:25am show on Monday 9th August, and you can buy your tickets on the Fringe website.  As well as some festival shows, Julia and Joel are going to have fun designing a new game called Mathopoly, based on the classic property game but with maths instead!  Watch this space to find out what they’ve created…