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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

Our own hands, and brains, had not been idle. We were actively looking for mathematical ideas to represent in a new way. We had played with geometric shapes but were now turning our attention to numbers. Counting was a bit too obvious and the next step seemed to be multiplication tables. Arranged above the picture rail in my classroom was a set of cards, each showing a different multiplication table with the multiples coloured in. They were useful for pupils to check on multiplication facts they might have forgotten. The coloured blocks formed a different pattern on each sheet. Colours, blocks and numbers combined to give us one of the best ideas we ever had.

We wanted to represent all the multiplication tables on one large grid. Finding a way to do it was a challenge. We knew it had to be a grid to represent the numbers one to one hundred. If we had been making it today even that decision might not have been so cut and dried as current mathematical thinking is tending towards using grids of zero to ninety-nine. We looked at the factors of all the numbers and realised there were more than we had thought about and we couldn’t possibly include them all. We could have colours to show that numbers will divide by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 but couldn’t go on to showing multiples of 11, 12 or anything higher. There were two reasons for this. One was that there simply wouldn’t be enough room to show them all. Secondly, and more importantly, the colours had to be easy to identify and anyone looking at the grid had to immediately know when they were looking at a particular colour, not left wondering if this shade of yellow was the same as that shade of yellow.

We needed the grid to be immediately obvious so an eleventh colour would be needed to separate the squares. Another decision was made but the planning was far from over. Some numbers only divide by themselves and 1, others have several factors. Once we had decided we would only include the factors up to 10, the largest number of factors in any number was seven. For instance, the number 60 will divide by 1, 2, 3, 4, 5, 6, and 10. Other numbers had one, two, three, four, five or six factors. The only way we could show these varying numbers, and for the squares to all look similar, was to use stripes. That didn’t make the task much easier because we still needed to decide how many rows we were going to have in each square to show all these different combinations of stripes. Eventually we had to settle for not having equally spaced stripes in every square but I don’t think anyone would ever notice if they hadn’t been told. The squares were 24 stitches wide with 24 ridges of garter stitch in each square. The squares that needed 2, 3, 4 or 6 stripes worked perfectly. The others had to have some stripes wider than others.

The knitting process was easy as the afghan was knitted in ten columns with the navy blue to separate the squares knitted as extra stripes on the way. However it did need a great deal of concentration to ensure that the colour sequence was correct. One big advantage of knitting small pieces, like the 24 stitch wide columns in this case, is that there is very little to pull undone when a mistake is made. When the columns were complete all that remained was to join the columns with navy blue separating strips and to add navy borders all round, including a channel for the hanging pole.

I remember the finishing of this more vividly than anything else I have ever made. When it all went together it was a shock. I had previously assumed I knew most of what one could ever know about the numbers up to one hundred. Ben was there at the time. He was already proving to be a very able mathematician with far more talent and knowledge than I ever had – and he could knit and crochet. We laid Counting Pane on the floor and stared at it. There was far more there than we had anticipated. We sat on the floor picking out one amazing fact after another. They came thick and fast.

What struck me first was the large number of colours in the tens column (10, 20, 30, etc.) I must have known it as I was knitting the column but it only jumped out when it took its place with the other columns. I have often been known to talk about ‘nice numbers’. To me, nice numbers are those that have lots of factors because they are easy to juggle with. I had never classed multiples of ten as nice numbers. They were ‘just tens’. Everyone knows you add a nought if you want to multiply by ten or take one off if you want to divide by ten. That didn’t make them interesting enough for my nice numbers. Now they were the first thing to shout at me. That column was far more colourful than any of the others. There were more factors than anywhere else.

It is obvious, when you stop to think about it, that any number that will divide by 4 also divides by 2. Everywhere there was a green stripe representing 4, there would be a yellow stripe for 2. Similarly, every square containing grey, for 8, must also have green and yellow. A brown 6 always went along with a yellow 2 and a red 3. Oranges, representing multiples of 9, made an interesting sloping line. We could have expected this because adding 9 is the same as adding 10 and taking away 1 so the squares are 1 away from lining up. We hadn’t expected the lines of red that seemed to criss-cross in all directions. Bright pink only appeared in two columns but, as it represented 5, perhaps that was not so surprising.

The other big surprise, that came to change my perception of certain numbers, was the number of plain blue squares there were. Theoretically, a square was blue if its only factors were one and the number itself. This is the classic definition of prime numbers. It is sometimes stated as ‘a prime is a number that has only two factors’. In Counting Pane not all prime numbers are blue so it cannot be used as a reliable definition. 2, 3, 5, and 7 are all prime numbers but they have a coloured stripe. However, what was very apparent was that the blue squares were clustered in certain columns. It should have been obvious that there would be no primes in the even-numbered columns, because 2 divides into all the numbers in them. Whenever I think about prime numbers now I think about them in terms of which column they are in.

Counting Pane had a huge impact on us and many other people who studied it later. It was, and still is, our pride and joy.  If we had to save just one it would be this.

Click here to see more about Counting Pane

9a. COUNTING PANE