Archive for the ‘Knitting’ 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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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!

Braided knitting

A few months ago a new sheep, Fernilee, appeared in my flat. (You may remember him from the New Year’s Eve party.) Luckily for him I’m not a very territorial sheep (except for the sofa, which is MINE) but I was a bit put out that he had a lovely hat and scarf and I didn’t. I know that it’s currently summer in Edinburgh, but that isn’t the point.

Seeing as Julia was totally bored after having finished her PhD, I asked if she would make me the scarf to rival all scarves. Not only should it be a warm and functional piece of winter knitwear, but it should embody some sort of mathematical principle so that I can continue inspiring my followers wherever I go. We brainstormed a few ideas: having a stripey scarf where the number of rows of each colour followed the digits of pi (e.g. see here), or having a hidden mathematical image knitted into the scarf (so-called illusion knitting), or having a braid pattern using cabling. The first idea seemed too easy and the second one quite hard, so we decided on a braid.

Braid pattern

The braid pattern we chose

Julia had never tried cabling before and wasn’t sure how to design a pattern from scratch, so we decided to find a ready-made pattern for our first braiding attempt. Eventually we decided to go with this one modulo some modifications – changing the border to seed stitch to make it easier, and adjusting the cabling pattern to make the braid alternate (over-under-over-under). It’s a 6-stranded braid with no special mathematical properties (that I can see). In particular, it is definitely not the same as this commutative braid which my officemate Patrick and my old supervisor Andrew are working with.

Being mathematicians (and knot theorists!) definitely helped us to figure out how the cabling pattern worked. In a mathematical braid there are a number of strands running parallel to each other, and every now and then two adjacent strands are allowed to cross. If the strands are labelled 1 to n, then the crossings are denoted by a sequence of numbers, where i means that strand i crosses OVER strand i+1, and –i means strand i crosses UNDER strand i+1. For example, the braid below would be denoted by (1,-2,1,-2):

Mathematical braid

The braid 1,-2,1,-2

This is in some ways very similar to knitting braids, because in a knitted braid only adjacent strands are allowed to cross. The cable pattern denotes whether the crossing is OVER or UNDER by using F (‘front’) and B (‘back’). The first obvious difference between the maths braids and the knitted braids is that knitted strand-crossings are allowed to occur simultaneously between non-interfering strands. E.g. strands 1 and 2 can cross at the same time as strands 3 and 4 do. Mathematically it makes no difference, but aesthetically it is more pleasing to have simultaneous strand crossings.

The next similarity between maths and knitting is how we ‘add’ braids together. It is simply by putting them side-by-side, the second braid following on from the first. Similarly, the knitting pattern only gives the first 16 rows – the first ‘block’ – and then the braid is continued by placing these blocks next to each other along the scarf.

group of robots

Braid theory can design paths for these robots so they don't crash into each other

Mathematically, braids are interesting because their addition has a lot in common with numbers. There is a ‘zero’ braid which does nothing when added to another braid – it is the braid with n strands running in parallel. There are also ‘inverse’ braids, which are like negative numbers in the sense that when you add a braid to its inverse you get the zero braid back. (Can you figure out the inverse braid to the (1,-2,1,-2) braid drawn above?) This additive structure makes the collection of braids into a group, and the braid group is of great interest to a lot of people in the world right now! Engineers use them for motion planning in robotics, cryptographers use them to design new codes and computer scientists are using them to design quantum computers.

Braids can also be turned into knots by joining the strands at the end of the braid back to the beginning. I think if I make another braided scarf I shall try to encode the braid for the knot 12a631, which is the only knot in my thesis where I couldn’t decide if it was slice or not.

I will end this post by showing you some pictures of how Fernilee reacted to my beautiful new scarf. You can see that he was quite jealous! 🙂

Fernilee looks on...

Fernilee looks on from behind as I try on my new scarf...

Fernilee looks on jealously

We have a little chat...

Inspection of scarf

He inspects the scarf and admires the mathematical braid

Ninja Haggis!

We make friends and play a game of Ninja Sheep!

If you’d like any more information or more photos of the scarf, please get in touch!