Archive for the ‘Mathematics’ Category

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!

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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: 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!

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!

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.

A Night in Wonderland

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

http://www.flickr.com/photos/peperico/4043195345/The theme for this month’s Museum Late was “A Night in Wonderland”, so there were lots of top hats, white rabbits and red queens! (See lots of photos of the event on the Museum’s Flickr page.) Knowing that Lewis Carroll (real name Charles Dodgson) was a mathematician and logician as well as nonsense-poem writer, it seemed wrong for there not to be a mathematical component to the evening so I got together with Madeleine Shepherd (from ICMS) to brainstorm some ideas…

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

FortunatusPurse Step 1

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

FortunatusPurse Step 2

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

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

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

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

Petersen Graph

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

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

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