## Giant 4D buckyball sculpture

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.

‘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

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: http://www.youtube.com/watch?v=VIVIegSt81k.

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

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 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. 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 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 http://www.snh.org.uk/publications/on-line/geology/scotland/default.asp or http://www.scottishgeology.com/geo/getting-started/.]

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…

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

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

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

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

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

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

The standard Borromean rings.

A different solution to the Valknut challenge.

And another solution!

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

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

## My new geek clock

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

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

1. ℏ × 1034 = 1.05 is the reduced Planck constant; that is, the Planck constant divided by 2π. This number relates the energy of a photon to its angular velocity and is one of the basic units in quantum mechanics. Its existence means that energy only comes in discrete packets, or quanta, which was a massive discovery in 20th century physics.
2. $\sum_{i=0}^{\infty} 2^{-n}$ 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).

## 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…

## Guest Post: Topological Crystallography in Stockholm

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

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

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

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

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

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

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

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

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

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

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