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The World of Illusion Knitting

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PICKING UP THREADS

This was written in
2007
so is now very dated

Strange things were also happening on the Maths front. Somehow I had been persuaded to run a workshop session at the following year’s Association of Teachers of Mathematics annual conference. I don’t know why I was doing it in my own name, instead of in partnership with Steve as we did in subsequent years. Maybe we hadn’t realised then that a session could be run by two (or more people). Anyway, it was in my name and I was terrified. Steve was there for moral support but, being a much less talkative person than I am, preferred to remain in the background. What we proposed to do at this session seemed extremely simple compared with some of the high-powered Maths going on elsewhere. I know we took a range of activities that we had evolved from our work with knitted shapes but I only remember one from that first year. We took cardboard and knitted squares, two cubes of plastic canvas, and some plastic shapes that would clip together to make geometric models.

When the delegates turned up for the session I was even more concerned. They included lecturers, advisors, researchers and others who would know far more than we did. The first task was something that we could have given to a three year old. Each person was given two squares split diagonally so that they were half red and half white. The problem was to decide how many ways they could be put together edge-to-edge to give just one red shape. The answer is two or three depending on definitions. Everyone agreed that they could make a triangle. Some said they could make two parallelograms while others argued that these were the same parallelogram reflected. The answer was unimportant. What mattered was the discussion and insight into the different possible interpretations of the problem. The rules had deliberately been left vague and it showed that most people make a snap decision and the answer may not always be so straightforward.

More half-and-half shapes were added and a great deal of fun was had as delegates twisted their heads or walked round their tables to decide whether they were making new shapes, reflections or rotations. There was much discussion about which should be allowed. Eventually, most being experienced in such tasks, they resorted to continuing the task systematically. Instead of scrapping all that had gone before, as children tend to do, they took the shapes they had found at the start and built on that knowledge adding extra squares in all orientations in all possible spaces.

They played together and had fun. There was one delegate who worked completely alone. It was a bit worrying because we knew him to be someone who was highly-respected in his field and we had given him this simple task. As the session we progressed we asked the delegates to draw what you see when you open up a cube and lay it flat. They could all do that. There are eleven possible answers and they’d all drawn them many times before. We gave them a cube with each of its faces a different colour and asked them to colour in their nets to match. It sounds easy – stand the cube on the table next to the net, see where it unfolds and colour the faces accordingly. But it’s not so simple. There are six different faces that could sit on the table. Each one could be four different ways round and they already knew that there were eleven ways of unfolding it. It proved to be a very nice task in visualisation and amazingly difficult for some of the more traditional mathematicians amongst them. Some resorted to using the plastic shapes so that they could fasten and unfasten a cube to check that they were right. There was another stage yet to come.

Our solitary worker was still working away on his own and had ignored the cubes completely. That wouldn’t surprise me now. It’s something I do myself at these events. The speaker says something that sets me off on a train of thought and I start to pursue it on the spot. Nobody minds but, as a newcomer to the situation, I was concerned that he was bored.

The rest started work on the next cube which had every face coloured half red and half white so that there were large red points at two of the corners of the cube. What would this look like when it was opened out? Those who had coped before began to struggle now and were very glad when coffee time arrived and they could take the problem away to sort out later. The other man did not go for coffee. He carried on working with his original squares. He had covered several sheets of paper with numbers and lists and columns. He had been devising an unambiguous form of notation to represent his findings and was fascinated and stimulated by this apparently simple task. It all served to prove that there are more different ways to tackle a problem than any one person can imagine.

It was probably at that same conference that the speaker at the opening session showed a black and white pattern and said, “Turn to the person next to you and say something about this pattern”. Steve was sitting on one side so I turned to the complete stranger sitting on the other side and said, “I could knit that!” Poor man. His face showed all that he was thinking! The pattern was made from the type of shapes we had been working with.  Of course I could knit it and I could find seven different ways to colour it in such a way that it was difficult to believe they were all versions of the same pattern. The colouring was easy and I sent the coloured versions to the speaker from that opening session. I believe he included them in an academic paper he was writing at the time. I did knit it. I knit it three times.

The knitting proved slightly more difficult. Had I been working on a flat surface it would not have been a problem but this was a sweater and the pattern had to wrap round the body and meet up. No matter how I tried I could not get it to match exactly. Sometimes, when things look as though they really should work, it is hard to grasp the reason why they just won’t fit. Fortunately, son Ben was on hand and spotted the problem instantly. “It’s irrational”, was his comment. At first I took this as a comment on my state of mind but then realised he was talking about irrational numbers.

Root 2 is one of those numbers with a long string of numbers after the decimal point that never ends. There is no exact answer. Knitting is very forgiving and the small discrepancies are taken up by slight stretching. The calculations would never work out but the knitting would look good enough to pretend they did. I knitted a lilac and white version, then an evening version in black with sparkly black shapes between. The third version was to be a long tunic with huge flowers, made from squares and rhombuses. It was impossible to arrange the flowers so they flowed correctly from back to front so we worked on the assumption that nobody could look at the back and front simultaneously, and that wearer would need to have their arms in the air for the cheating to be revealed. So far no one has ever mentioned that some flowers are missing a petal.

Click here to see ways of colouring the design

7b. AROUND THE WORLD continued