The Imitation Game: Part 3

In The Imitation Game: Part 2 we looked at Alan Turing’s work doing codebreaking at Bletchley Park during World War 2. In this final instalment of blog posts about the film The Imitation Game, we look at the work Turing did during the final years of his life, about pattern formation. This is based on a talk given by Professor Jamie Davies from the University of Edinburgh at an event called The Maths of the Imitation Game at the Filmhouse cinema.

You might be fooled into thinking that the film doesn’t touch on Turing’s work after Bletchley at all, but you would be wrong. The reference is subtle, but it is there from the very first scene. The opening part of the film shows Turing in his house, clearing up mess from a burglary. On a table we see a pine cone; on the walls we see pictures of strange spirals of dots and curious pictures of starfish. These pictures are again present in the final, very moving, scenes of the film, when Turing is at home suffering from the effects of his enforced chemical castration. But what do these images tell us about the work that Turing did?

pinecone with spirals

Count the number of spirals on a pinecone (first four shown here). You often find a Fibonacci number.

Count the number of petals in a daisy, or the number of spirals in a pinecone, or the spirals in the head of a sunflower. You will very often get a Fibonacci number: one that appears in the sequence 1,1,2,3,5,8,13,21,34,… In this sequence, you get the next number by adding up the previous two.

Turing claimed that the ubiquity of Fibonacci numbers in plant forms was not a coincidence, and wanted to show that such patterns could arise as the result of chemical reactions in the cells of the plant. In this view he was influenced by the book On Growth and Form by the Scottish mathematical biologist D’Arcy Wentworth Thompson.

In 1952 Turing wrote the paper “The chemical basis of morphogenesis” which was ground-breaking work in explaining how local chemical reactions could result in large-scale patterns. Fibonacci numbers in pine cones, stripes on a tiger, tentacle patterns on a starfish and hexagons in a fly’s eye were all predicted to be manifestations of the same type of chemical reaction. Today Turing’s idea is called a reaction-diffusion equation and is used in even more areas of science than just animal biology.

Haeckel's Kunstformen der Natur

Haeckel‘s picture showing various sea anemones with their tentacles. If you look closely in the film you can spot this picture on the wall of Turing’s house.

The idea is that patterns are formed by the interplay between two types of chemical: one that activates growth (called the activator) and one that inhibits growth (called the inhibitor). Production of the activator stimulates further production of the activator, but also stimulates creation of the inhibitor. Imagine a ring of cells around an embryo. A particular cell might grow ever-so-slightly faster than its neighbours, resulting in more of the activator being produced and thus even further growth. At the same time, the cell is producing the inhibitor chemical, meaning that the growth of cells adjacent to it is stunted. However, cells that are far enough away are not affected by the inhibitor, and so will grow as normal. This results in a sort of wave-like pattern around the embryo, of particular cells growing at regular intervals. And this is exactly what causes the regular arms of a starfish or the tentacles of a sea-anemone.

Turing was not only the first to come up with these ideas, but the first to test them using computer simulations. Working at the University of Manchester in 1951, he had access to the world’s first commercially available general-purpose digital computer, the Ferrati Mark 1, and he immediately started programming it to explore the consequences of his equations. Working by hand his equations would never have been tractable to solve, but using a computer he could see that they were a great model for the physical phenomena he was trying to explain.

Sadly Turing’s work was far ahead of his time, too difficult to understand for biologists (even though he made a conscious effort to explain things well!), and was largely ignored by the scientific community for fifty years. It was only in the late 1990s that the work was picked up again and Turing’s paper rediscovered, and today reaction-diffusion models are considered fundamental to the study of pattern-formation. Prof. Jamie Davies’ current research in Edinburgh is testing Turing’s ideas by ‘programming’ real living cells to make patterns on command, and his lab has been able to replicate all the predictions made by Turing.

To conclude, I think it is a shame that the film portrays Turing as being this incredibly autistic and single-minded person, giving all his love to machines and algorithms instead of real people. In reality he was a warm and friendly individual who had strong friendships and relationships, and, moreover, he did some of his most important work into living things rather than cold machines.

I hope my series of blog posts have highlighted the amazing research that Turing did, which revolutionised computer science, mathematics and biology, not to mention the pivotal role he played in breaking Enigma during World War 2. Many thanks go to John Longley, Tom Leinster and Jamie Davies for providing the material on which these posts have been based, and to the Filmhouse cinema in Edinburgh for hosting our event explaining the Maths of the Imitation Game.


The Imitation Game: Part 2

In The Imitation Game: Part 1 we looked at the early work of Alan Turing about universal computing machines, the limits of what computers can do, and whether computers could ever successfully imitate human brains. In this post we look at Turing’s work doing codebreaking at Bletchley Park during the Second World War, and the similarities and differences with codebreaking today. The material in this post is based on a talk given by Dr Tom Leinster from the School of Mathematics at the University of Edinburgh, as part of an event called The Maths of the Imitation Game at the Filmhouse cinema.

Bletchley Park in Buckinghamshire was the site of the Allied codebreakers in World War 2.

Bletchley Park in Buckinghamshire was the site of the Allied codebreakers in World War 2.

Bletchley Park, otherwise known as Station X, was a mansion in Milton Keynes that was used during World War 2 as the base for the Government Code and and Cypher School (GC&CS). (Earlier this year it was reopened as a museum, and I’m told it’s very good!) One of its main purposes was to decipher German messages, which were being encrypted using a device called the Enigma machine. It was thought that Enigma was unbreakable, and so it might have been were it not for Turing’s brilliance and subtle mistakes made by the German operators.

In a simple substitution cipher, each letter of the alphabet is replaced by another letter of the alphabet. For example, if we had S->F, H->U, E->R and P->Y, the word SHEEP would be encoded as FURRY. Such a code is easily breakable using frequency analysis: if E is the most common letter in the English language and R is the most common letter in the message, it’s likely that E has been encoded as R. The Enigma machine is much cleverer than that.

A 3-rotor Enigma machine with parts labelled.

A 3-rotor Enigma machine with parts labelled.

In a standard Enigma machine there are 3 rotors. Each rotor has each of the 26 letters of the alphabet inscribed around it, and each is set to an arbitrary position at the beginning of the day. When the operator types a letter on the keyboard, the signal from that key is sent through wiring to the first rotor, which encodes it as a new letter. It is then sent through wiring to the second rotor, which changes it again, and finally it is sent to the third rotor, which changes it again. The signal then goes around some fixed wiring (called a reflector) and then it returns through each of the rotors, finally lighting up a new letter on the lampboard which the operator writes down. Here’s the clever bit: after each new letter is entered, the rotors turn. Therefore, if you had typed S the first time and got an F, you could type S again and get a Z. The same letter is encoded differently every time it is pressed. The message itself is part of the encoding.

There are 3 different rotors, which the operator could choose out of a possible 5. This gives 5x4x3=60 options already for the initial setup. Then, each rotor can start out in any one of 26 different positions. This gives 26x26x26=17,576 options. So far, this is only about a million combinations. It sounds like a lot, but this is at a level where, with a little ingenuity, you could simply brute force all the possibilities. To make Enigma unbreakable a final layer of encryption was added: the plugboard. This used cables to pair up letters. For example, if E was paired with Q, then if E was typed then the machine would interpret this as a Q before transmitting it through the rotors. With this additional scrambling, the number of combinations of an Engima machine was over 159 million million million. There was no way they could ever hope to check all possible settings in a reasonable amount of time. And, to make matters worse, the Germans changed their settings each day, meaning that there was only 24 hours to figure out each code before everything changed.

What Turing did to crack Enigma was to build a machine capable of doing logical calculations that would eliminate a vast number of the possible settings. This was called the bombe and built on earlier work by Polish cryptanalysists. Basically it could try different rotor settings in turn, and look for logical contradictions that would show the settings to be impossible. (For an analogy, think of doing a sudoku puzzle where you might postulate that a 6 goes in a box, but that would result in two 3s in another row which can’t happen, so therefore it can’t be a 6.) Such contradictions might include:

  • Deciding that a letter was encoded as itself. This was impossible due to the way the signals through Enigma were sent round the reflector.
  • Having an asymmetry in the plugboard. If the B is connected to the N, then the N must also be connected to the B.
  • Having a letter in the plugboard connected to its neighbour. Operators were told not to do this.
  • Having plugboard settings that were used the previous day. Operators had to change all the settings every day.

There were many more contradictions like these. There were also sloppy practices among the operators. They would often not set the rotors to a truly random initial position, but would use their names (e.g. “BOB”) or would simply turn the rotors a few places from the day before. There were also common phrases in messages, e.g. the word “ein” appeared in 90% of messages, and “Heil Hitler” also appeared often. Wikipedia has a fascinating description of the methods used to crack Enigma, as well as the methods the Germans used to make Enigma even harder to crack.

Nowadays, encryption of information is done using mathematical algorithms rather than mechanical machines. The most commonly used is called RSA, and relies on the difficulty of factorising large numbers into primes. Just as in Turing’s time, codebreakers rely on having a large amount of information to help them look for patterns, and on computers to do the decryption. The big difference between then and now, according to Tom Leinster, is that during the war the government was spying on the Nazis; today they are spying on us.

Edward Snowden

Edward Snowden, who revealed details about the mass surveillance carried out by the NSA and GCHQ.

Earlier this year it was revealed that the UK’s Government Communications Headquarters (GCHQ) and the USA’s National Security Agency (NSA) had been systematically monitoring all of our emails, phonecalls, texts, web browsing and bank transactions. Their goal was to “collect all of the signals, all of the time”, regardless of whether or not anybody had done anything suspicious. And, just as in the film, the codebreakers often had complete autonomy over the information they collected, with even the highest in command being unaware of what they were doing.

Just as in the film, this codebreaking was only possible because of errors in the encryption of the data. The information leaked by Snowdon showed that the NSA had inserted a secret back door into the world’s most widely used cryptosystem, allowing it to break the encryption.

Alan Turing was a homosexual at a time when homosexuality was illegal, and his conviction and subsequent chemical castration were what led to his suicide in 1954. Today it is legal in the UK for two men to have sex, and even get married. This change in our law has come about because of campaigning and activism, but it is always dangerous to be an activist, speaking out for something that is considered against the law. It is easily argued that such campaigning is even more dangerous today, with the government carrying out mass surveillance of everyone in the population.

I shall end this post with the question asked by Tom Leinster in a piece he wrote for New Scientist in April: is it ethical for mathematicians to work for government intelligence agencies like GCHQ?

I look forward to hearing your comments!

Read The Imitation Game: Part 3 about Turing’s work in biology and pattern-formation.

The Imitation Game: Part 1

Imitation Game posterToday we had a wonderful event at the Filmhouse cinema in Edinburgh talking about The Maths of the Imitation Game. This is the film which tells the story of mathematician Alan Turing and his work codebreaking at Bletchley Park during the Second World War.

I’m going to admit up front: I liked the film. I’ve met other people who’ve been foaming at the mouth with anger over inaccuracies in the film, both with the historical aspects and also with how Turing was portrayed. I agree that the film isn’t perfect – they had to take a lot of liberties with how things were presented in order to make the story appeal to a mass audience. Events certainly did not happen the way the film depicts. And I do take issue with the film’s insinuation that Turing assisted Soviet spying, and also that Turing would have told classified secrets to a police officer. But I also think the film gets a lot of things right, it tries its best to explain key aspects of the story, and is wonderfully acted, especially by Cumberbatch as Turing.

If nothing else, the film is a great platform to start discussions about the important (theoretical) work that Turing did in his life and the repercussions of that work today. This is what our event at the Filmhouse was all about. We had three speakers talking about different aspects of his work, and I’ll summarise those discussions here. I’ll break it down into three blog posts as there is a lot to say about each one.

First up, Dr John Longley from the School of Informatics talked about Turing’s early work on computability. In our lives we are used to having different machines for different tasks: a toaster that warms your bread, a dishwasher to wash your plates and a blender to mash your food. We wouldn’t expect a single machine to perform all these different tasks. Yet, walk through to your study and you’ll find a machine on which you can do many things: play solitaire, manage your accounts, watch videos, search for prime numbers. Nowadays we aren’t at all surprised that a computer can do so much, but in Turing’s day such a machine was inconceivable. In fact, it was Turing who proposed that such a machine could exist. He had the idea of a ‘universal machine‘ which could emulate the output of any other given machine.

Every machine takes some sort of input, performs a certain task and then gives an output. The toaster’s input is your bread, which is heated according to the settings you’ve told it, and its output is some toasted bread. The Enigma machine takes as input the key presses of letters, it performs an encryption, and it outputs letters forming a coded message. For Turing’s universal machine, the input is a sequence of symbols that provide the instructions for what another simulated machine does and the input to this machine. The output is then the answer that the simulated machine would have provided if you had given it this particular input. This idea that the input may itself comprise a set of instructions was groundbreaking. Today we simply call this kind of input ‘software’.

Turing’s invention of the universal machine was designed to answer a long-standing problem in mathematics: can the answer to every mathematical question be determined mechanically? This was posed by David Hilbert in 1928. When Turing was only 23 years old, he showed that the answer to this question was negative, and he did this by coming up with something called the Halting problem.

Turing's bombe

Will Turing’s machine output the correct Enigma settings, or could it keep searching forever? This is called the Halting problem.

This simply asks: given a particular input to a machine, is it possible to determine whether the machine will ever halt and give an output? For example, in the film the machine (called “Christopher”) that Turing has invented is checking through possible settings of Enigma and everyone is waiting for it to tell them what the correct answer is. Will it ever stop, or will it keep whirring away for ever, stuck in some logical loop that prevents it from finishing? If you pull the plug, how can you be sure that in the next minute it wouldn’t have told you the answer? Turing proved that there is no single algorithm that can decide whether a given arbitrary program will halt for a given input. The halting problem is ‘undecidable’.

The conclusion from his work is that there are some questions in mathematics that are unknowable –  there is no computer program that will ever tell us whether they are true or false. However, this begs the question: is the human mind the same as a computer? Or could a brain do computations, do mathematics, that a machine could not? If so, what makes a brain different from a computer?

This question fascinated Turing and led him to invent The Imitation Game, which gives the film its title. Rather than asking the question “Can machines think?” which is very abstract and hard to pin down, he instead asked whether it was possible for a machine to imitate a human so well that another human could not tell that it was really a machine. In a Turing test, person A can interrogate player B by asking a series of questions. Turing said that if the interrogator decides as often as not that player B is a real person, then the machine has passed the test. Despite predicting that computers would be built to pass the test by the year 2000, it was only in June this year (2014) that a computer was said to have passed the test. This was a computer masquerading as a 13 year old Ukrainian boy, who fooled 33% of a panel of judges over the course of a 5-minute conversation.

What do you think? Could computers come to imitate humans one day? Is it just a case of having more computing power, or is it possible that brains can do things that computers fundamentally cannot?

Many thanks to John Longley for the material on which this blog post is based.

Read The Imitation Game: Part 2 about Turing’s role in codebreaking during World War 2 and moral questions about codebreaking today…

Orkney and beyond

I used to believe that planes always landed on runways.

Orkney has a way of stopping you from taking things for granted.

oisf-logoI was up to speak for the second time at the Orkney International Science Festival, which is organised by Howie Firth – one of the most enthusiastic men I have ever met. He has a way of making you feel that each thing you say is the most interesting thing he’s ever heard. So it was with his usual infectious enthusiasm that I was invited up to speak about Botanica Mathematica and the links between maths and knitting.

With true Orcadian hospitality, Howie’s invitation didn’t mean that I came up to give my talk and then had to leave immediately after, but was an opportunity to have a holiday and time to explore the islands. Last year my companion Albert and I investigated the Mainland, seeing the amazing neolithic site of Skara Brae (the best-preserved prehistoric site I’ve ever seen), the stone circles of Brodgar and Stenness and the amazing coastline at Yesnaby. This year, it was time to venture further afield…

Orkney Map with North RonaldsayThe weather forecast had promised an overcast but dry and mild day for flying to North Ronaldsay. Nothing could have been further from the truth. Morning broke to gale force winds and torrential rain, neither of which eased up for the entire day. Apparently a storm system had come in from the north east, bringing vengeance on Orkney and Shetland but leaving the rest of the UK to enjoy beautiful warm sunshine. Sigh.

To say that I was scared of the impending flight was an understatement. It was basically a flying minibus – notionally with 9 seats, but one of those seats being next to the pilot. The pilot in our case was Rebecca Simpson, a cheerful blonde woman of about 30 , who seemed amused at the terrified looks on our faces. We had a 30-second safety briefing, were told to buckle our seatbelts and then the propellers went to full throttle.

I can easily say that the flights that day were the best I have ever been on. The plane needed hardly any runway before it was in the air, buffeted by the winds and quickly gaining height to give us a spectacular view of the azure blue of Orkney’s various harbours. Our first stop was Papa Westray, which is mainly famous for having the shortest scheduled flight in the world – less than 2 minutes over to the neighbouring island of Westray – which comes with its own certificate.

The "airport" at Papa Westray

The “airport” at Papa Westray

Despite my lack of certificate, I was glad that I was on the longer flight from Kirkwall, with time to enjoy the views and the feel the force of the weather blowing us around. Our landing on Papa Westray really showed off Rebecca’s skill; the winds forcing us to approach the runway facing about 45 degrees away from it, but turning at just the last moment to achieve a perfect landing. I was also incredibly amused at Papa Westray’s “airport” – bascially just someone’s house.

Five minutes later we had landed on North Ronaldsay, and were gratefully met by Tommy Muir, who was going to give us a tour of the island. Our original intention was to have a day of hiking about the island, but the weather meant that we didn’t want to be outside for more than a few minutes at a time, and were glad of the shelter of his van!

(C) Lis Burke

Seaweed eating sheep

North Ronaldsay is about 3 miles long and is mainly famed for two things: having the tallest land-based lighthouse in the UK, and for having seaweed eating sheep. In 1832 a dyke was built around the island and the native sheep were exiled there to make space on the island for more lucrative breeds of sheep and cow. The hardy creatures learnt how to survive on the seaweed and became renowned for their resilience, intelligence, tasty meat and soft wool. (Indeed, few sheep breeds have their own sheep fellowship!)

North Ronaldsay once had as many as 500 people living on it; today there are no more than 50. Climate change has meant that the land is no longer suitable for growing crops on, and so people have left as they realise there is no work for them to do. There is a school there, but only one child to attend it – teachers are flown in from the mainland to provide art, sports and history lessons. Some tourists do come, seeking the tranquility and remoteness of the place, and often to watch the seals and birds on the coast. Last year there was apparently a walrus who visited the island!

Despite a wet and windy day, we were sad to leave and were determined to visit again on a sunnier day.

Me with our amazing pilot Rebecca

Me with our amazing pilot Rebecca Simpson

Rebecca was there with her plane to take us home, and this time there was a dog occupying one of seats! He seemed completely nonplussed by the turbulence of the plane – he’d probably been on more flights in his life than me! Our stop in Sanday on the way home was another adventure. The direction of the wind made landing on the runway very difficult, so Rebecca simply landed at right-angles to the runway, into a field instead! She seemed to love the challenge of the weather conditions, but told us afterwards that the winds were quite mild compared to what she’d had to deal with before.

Back in Kirkwall airport, the giant runway with all its lights seemed far too easy for Rebecca, and we knew that no flight we ever took would be quite as exciting again. My talk on Monday night was well received and I’m hopeful of getting some new binary bonsais and hyperbolic chanterelles to add to our collection. The hospitality and enthusiasm of everyone I’ve met in Orkney has meant that I will no doubt be back for many years to come, always finding a new adventure and wonders to explore.

And, if this story has inspired you to visit Orkney and talk about science, get in touch with Howie and he’ll no doubt be eager to have you visit to speak at his science festival!

Giant 4D buckyball sculpture

4D buckyball Zome sculpture (c) Graeme Taylor

This is a model of a mathematical structure called a “Cantitruncated 600-cell”, colloquially known as a 4D buckyball. It took twenty people five hours to build and contains over 10,000 pieces of specialised plastic called Zometool. Such a model has never been seen in the UK before and I’m incredibly proud to have been able to organise its creation in Edinburgh last week.

The sculpture perches at the top of the main staircase in Summerhall, a great arts venue which used to be the University of Edinburgh’s veterinary school. The hall in which we put together those pieces of plastic was no doubt designed for dissecting cows or lecturing students about the removal of dogs’ testicles. Instead, Monday’s event (held as part of the University’s Innovative Learning Week) led our students into looking at the anatomy of geometry and playing with very different sorts of balls.

So what is a “Cantitruncated 600-cell”? The description on Wikipedia is less than enlightening. (It does, however, give some other cool names for this shape, including the “Cantitruncated polydodecahedron” and “Great rhombated hexacosichoron“.) Basically, the 600-cell is a shape made up of 600 tetrahedra (which in turn are 3D shapes made of 4 equilateral triangles) joined so that 20 of them meet at each corner. To ‘truncate’ means to ‘chop off the corners’. If we chop off a corner of the 600-cell, we see a shape which has 20 triangular sides – this is another regular 3D shape called an icosahedron.

Chop corners off an icosahedron, and you get a football, or buckyball.

Chop corners off an icosahedron, and you get a football, or buckyball.

‘Cantitruncation’ means ‘truncate, then truncate again’. Truncating the icosahedron leaves us with a shape colloquially known as a buckyball, or football (see left). Putting these facts together, we see that our model is a 4D shape made of 600 tetrahedra, but where each corner has been chopped off and replaced by a buckyball.

I have written a lengthier and much better explanation of this for the School of Mathematics website so recommend that you read that for more details! Otherwise just let your brain gently simmer in the crazy complexities of 4-dimensional geometry.

Photographer (and mathematician) Graeme Taylor was there on the day to do time-lapse photography of the build, and you can watch his final video at:

You can also see photos on Flickr by the University’s photographer Dong Ning Deng (scroll right for more!). Our students had to work very hard to not only put this giant jigsaw together, but also to cope with the engineering challenge of building enough of a framework to not let the model collapse under its own weight. I have to say that the sound of cracking plastic haunted my dreams for some nights afterwards…

Our 4D buckyball will stay in Summerhall until the end of the Edinburgh International Science Festival (20 April) and will (hopefully!) form part of the festival’s Art Trail. So go and see it while it’s there and tell me what you think of it!

Woolly toys

Maths knitting by Pat Ashforth

Pat’s knitting display at MathsJam

My flatmate Julia has been busy these last couple of months, knitting and crocheting mathematical toys for me to play with. Her inspiration came from meeting Pat Ashforth at last year’s MathsJam. Pat and her husband Steve are the authors of the wonderful website Woolly Thoughts, which contains patterns for all sorts of knitted mathematical wonders. Blankets, cushions, hats, scarves, puzzles,… All of which are guaranteed to bring smiles to the friends, family or colleagues that you show your creations to!

The first thing that Julia decided to make was a flexagon cushion. A flexagon is traditionally made by folding a piece of paper into triangles (or squares) which then folds into a hexagon (or a bigger square) and can be ‘flexed’ to reveal hidden sides to the shape. It’s difficult to describe in words! I suggest you download a flexagon template and get folding – you will soon be hooked on the idea. The advantage of having a crocheted hexaflexagon is that it’s very robust and can’t be torn by playing with it too much. It’s also easier to unwind it a bit and see the structure of how it fits together. It turns out that a hexaflexagon is just a 3-twisted Möbius strip!

Here’s a short video of Julia playing with the hexaflexacushion:

Can you track all the different colours?

Of course, no education on hexaflexagons would be complete without watching the wonderful videos by Vi Hart, including a Hexaflexagon Safety Guide. See the first of them here:

The second toy that Julia made is called an Octopush. This can be confusing if you google for it, because it’s also the name of an underwater sport. The toy is made of 8 cubes sewn together into a 2x2x2 mega-cube, and the colours are such that it is possible to flex the cube into lots of other positions. As with the flexagon, this is much easier to describe by showing you the video:

I’m not particularly impressed with Julia’s first attempt at knitting this, as the cubes aren’t perfectly cubical and it doesn’t fit together very neatly. But I guess we can’t expect humans to get it right every time. Hopefully she’ll make a better one someday. Can you figure out how it all fits together?

So, what should I get Julia to make next? Suggestions welcome!

Holiday in the Highlands

Photo by Floris Boerwinkel (a great name!)

Where do you think this is?

Palm trees, pristine white beaches, turquoise blue water…and sheep. If such a scenario sounds like your idea of heaven, you need to get yourself to the western highlands of Scotland. Yes, Scotland. And no, I haven’t embarked on an alternative career as a travel agent – I’ve just had a fantastic holiday up there and and finding it difficult to keep my enthusiasm to myself.

Well, I say ‘just’, but I’ve been back for weeks now. It’s taken me this long to adjust to being around people again. One of the great things about the Highlands is that there are more sheep than people. Not only that, but these sheep are exceptionally brave, talented and heroic. They will race down mountains at a 60 degree angle. They will climb over intricate rock formations in search of the tastiest seaweed on the beach. And, if they so wish, they will stand in the middle of the road regardless of whether any human-driven vehicle is racing towards them. Usually just around a sharp bend. Incredible. I wish I had such bravery sometimes. You should have seen my terror at the simple prospect of needing a shower at the end of my holiday.

But enough about me and my brethren. The north-west of Scotland has some of the most awe-inspiring and beautiful countryside I’ve seen in the UK, combined with magnificent rock formations and crazy geological phenomena. If possible, take a geologist with you on your travels, as I did, so that you can enjoy their geo-erotic tales of cleavage, orogenies and thrusts.

Albert and Treebor in the car

Albert dozes while Treebor enjoys the journey

My travelling companion, Albert, is officially a chemist but is secretly a wannabe geologist. He didn’t have so much to say at the beginning of the trip, as we travelled north out of Edinburgh, through the Cairngorms, past Inverness and then north-west to Ullapool. The first bit of excitement we had was as we turned off the A835 onto a single-track lane (the first of many!) towards Loch Lurgainn. Out of nowhere popped two surprising things: the mountain Stac Pollaidh, and a stowaway passenger called Treebor!

Treebor was only a month old, having been born as part of the Botanica Mathematica project to knit/crochet mathematical plant forms. He is what we call a binary tree, with his branches ever splitting off into two.  He had hidden away in our car, desperate to explore the great outdoors with us, despite being told he was too young. Before we could stop him he was racing up a mountain and hiding in the long grass – can you spot him in the picture below?!

Treebor runs up Stac Pollaidh in the long grass

Treebor runs up Stac Pollaidh in the long grass. Can you see him?

As I say, Stac Pollaidh (pronounced “Stack Polly”) is a surprising mountain, rising by itself out of nowhere from the surrounding landscape. Its peak is eroded in a very distinctive way as a result of being above the ice during the last ice age. This makes it an example of a ‘nunatak’ (presumably pronounced “nun attack”) and a favourite with climbers. It took us so long to find Treebor (have you found him yet?) that we abandoned walking all the way up the mountain and instead enjoyed the view for a while. After lunch on a (windy!) beach at Reiff, overlooking the Summer Isles, and a coffee at a wee pub in Altandhu, we drove north again past Loch Ra and Loch Vatachan (which sound like two evil nemeses!) and walked to the Inverkirkaig Falls, from where another impressive nunatak, Suilven, can be seen. Suilven dominates the local landscape, overlooking the town of Lochinver where we stayed the night.

Old Man of Stoer

Old Man of Stoer rock formation

After a good night’s sleep, we continued the drive north, along a single track road with blind summits, crazy bends, kamikaze sheep and amazing views over white beaches. Albert also has a bit of a thing for lighthouses, so we made the pilgrimmage to Stoer Head Lighthouse which was built in 1870 by the Stevensons. More interesting to me was the walk along the coast to a rock in the sea called the “Old Man of Stoer”. To me, it looked quite like a constipated monkey’s head than an old man. Why is there never a rock that looks like a sheep’s head?

Heading back to Ullapool along the A837, the geologist travelling with you will never forgive you if you don’t stop at the visitor centre at  Knockan Crag, the scene of one of the greatest discoveries in geology! A logical assumption in geology is that rocks are laid down in sequence with the youngest at the top and the oldest at the bottom. The puzzle at Knockan Crag though, was that the oldest rocks were on top and this lead to a heated argument…

It took two Scottish scientists, John Horne and Benjamin Peach, to (a) stand by their belief in the ages of the rocks, whilst many others tried to prove they had got it wrong, and most importantly (b) explain how the older rocks could have come to be above the younger ones. The answer was a previously unknown structure called a Thrust Fault. In this case the Moine Thrust, which was formed during the collision of two continents, when one sheet of rock was pushed over the other, kind of like a pile of paper being pushed over another pile.

At the visitor centre even more amazing facts were revealed to me:
– Once upon a time Scotland lived near the south pole, moving up to the equator and becoming a baking desert for a while.
– During this time it was part of a continent with Greenland and North America.
– Scotland and England collided about 400 million years ago, making the original ‘union’ of the countries. The forces involved created mountains like Ben Nevis and the Cairngorms and created the Moine Thrust.
– Around 60 million years ago the Atlantic started to be created, pushing Scotland eastwards and creating many volcanoes, like those on the Isles of Skye, Rhum and Mull.
– When the North Sea was created, Scotland very nearly broke away from England, but Scandanavia got pushed aside instead. (Our one true chance of independence and we blew it!)

[For more on Scottish geological history, visit or]

Bedtime stories don’t get much better than that! Not being able to tear Albert away from the Moine Thrust, we followed it all the way down to Skye over the next few days of the holiday. I insisted on making a detour to the wonderfully named “Isle of Ewe”, although this caused some consternation when I had to break it to Albert that I didn’t actually (love him). Skye is another place that is full of a huge variety of landscapes, from the Red and Black Cuillin mountains, to the majestic rock formations of the Quirang range and the tall cliffs out at Neist point where, of course, there was another Stevenson lighthouse we had to visit. It was great for young Treebor with his short attention span.


I said there would be palm trees, right?

I must also not forget to recommend the picturesque village of Plockton, which is well worth spending the night at on your way to Skye. Not only has it got a great name, palm trees, a beautiful loch and friendly people, but it has the best restaurant we ate at all week: Plockton Shores. Seriously, go there. Nuff said.

Sigh. Remembering my holiday I feel sad once again to be back in a busy city, which feels like a completely different country to the remote majesty of those mountains and lochs. Hopefully I will be back soon. In the meantime, I am busy getting on with a few knitting projects – stay tuned to find out more!

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…

Viking Museum Late

(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.

Valknut on Viking stone

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. 🙂

Standard Borromean rings

The standard Borromean rings.

Non-standard Borromean rings

A different solution to the Valknut challenge.

Non-standard Borromean rings 2

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.

Geek clock

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. \sum_{i=0}^{\infty} 2^{-n} is the sum 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots 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. \int_1^5 \ln x \, dx = 4.047 is the integral (area under the curve) of the natural logarithm function between x=1 and x=5. 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, i is equal to e^{-\frac{\pi}{2}}.
    7. \phi^{2^2} = 6.84 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. NA.kB = 8.314 is the ideal gas constant which relates the energy of a mole of particles to its temperature. It is the product of NA, which is Avogadro’s constant (the number of particles in one mole of a substance – see 12), and kB, which is the Boltzmann constant (which relates energy of individual particles to temperature).
    9. me × 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. \frac{ \pi^e}{2} = 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 \pi^e is transcendental, although we know that e^{\pi} 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!

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…