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On the day I visited the Science Museum to hand over Square Deal, Counting Pane and Pythagoras Tree, the Curator of Mathematics, Jane Wess asked, ‘Can you knit a Peano Curve?’ I had no idea as I had never heard of a Peano Curve. She said she would let us have some information and a few days later we received a letter containing some pages photocopied from an academic tome. We studied it but decided it could not be done.

We learned that a Peano Curve is a space filling curve and is rather like a fractal. It follows a simple path round a square but each section of it can be split up to reproduce that path on a smaller scale. Theoretically this happens again and again until the whole surface is completely covered. We couldn’t knit it but we kept the paperwork.

About two years later, on our way home from a maths conference, we went to visit Alec Dalglish of the Knitting Craft Group who we had been communicating with him for some time. He was retiring and would have liked us to take over the running of the organisation. It wasn’t for us. We knew it wasn’t the way we wanted to go but, as our journey home from the conference took us close to where he was, we agreed to visit him. We spent a delightful afternoon in his office. He produced interesting box after interesting box and it was all very tempting. He had several items created by well-known knitters and crocheters including work by Jan Messent, James Walters and Sylvia Cosh. We already knew these people and had often seen examples of their work. We could have taken charge of all those wonderful creations. All the pieces were far more artistic than anything we had ever made. It was an interesting opportunity but clearly wasn’t our path. Nevertheless, something amongst that treasure trove triggered something in my brain. It was difficult to concentrate on what he was saying as a brilliant idea had forced itself to the front of my brain.

Steve knew something was going on in my head. He’d seen me go off at a tangent often enough before. This was more than a tangent. It was time to abandon all our self-imposed rules for shapes and go in a totally different direction. I had not been able to knit a Peano curve but I could crochet one!

A Peano curve is a complex thing. It twists and turns and packs itself together so that, theoretically, it fills all the space on a surface. The computer, and son Ben, came to the rescue again. Ben has the ability to assimilate information more quickly than either of us and in the time we were thinking about how best to draw it he had drawn five or six generations of the curve showing how it packs itself together. To turn this curve into something we could hang on the wall needed a bit of thought and we eventually decided it had to be in two parts. First it needed a grid to represent the graph paper or squared paper it would normally be drawn on. The curve itself could then sit on top of this.

I am not a very experienced crocheter but this would only require basic skills. The first problem was how to get perfect squares in the grid, with the spacing needed. I made several attempts and then thought of a way to avoid some of the trial and error. By the magic of email I had a reply from James Walters within the hour. James is an expert crocheter, with a thorough understanding of the maths needed, and he had the information at his finger tips.

I made the grid, in black, with perfect squares. The curve was in the brightest shade of green I could find. The technical part of the crocheting was simple. It was just a chain attached to the grid at strategic points. The demanding part was keeping track of those strategic points. Originally I was trying to work from a single sheet of paper showing the entire curve. When it is finished it is clear that the curve forms little blocks, and lines can be seen criss-crossing the surface between these blocks. It is extremely logical and systematic but it was impossible to keep track of what  went where. Printing the four quarters on four pieces of paper helped a little and when those sheets were folded again for each small section of the curve it became considerably easier. When it was finished it was given a black backing so the grid lines could scarcely be seen and the green curve made a very dramatic squiggle across the black, going in at the bottom left corner and leaving again at the bottom right corner. This was a new technique, for us, and, if I was starting it again now, I would do it differently. The original is functional but could have been more ‘artistic’.

Just after it was finished we allowed ourselves to be talked into going to a Maths and Art weekend at Oxford University. We didn’t want to go. We were sure we would be out of our depth. We didn’t know enough about maths or art to mix with such august company. Against our better judgement, John Sharp talked us into it and asked us to bring the newly-completed hanging. We went. We took the hanging. I also took a cute cuddly teddy bear. Maybe I thought appearing to be really eccentric would be an advantage and besides he was a rather special bear. He was dressed in a Mobius scarf and waistcoat, both of which have strange properties as they have only one side, instead of an inside and outside as you would normally expect.

This first crochet hanging was named Peano Beano. Elements of doubt began to creep in when John whispered that the name was actually pronounced ‘pee-arno’, so the name made little sense. We had not had time to assimilate this information when it got worse. One of the professionals announced that it was a Hilbert Curve, not a Peano Curve. We hadn’t heard of a Hilbert Curve so we were in no position to justify our claim. Before we had chance to collect our thoughts, we were witnessing a full-scale argument. There were two very distinct camps; one lot favouring Hilbert, the other Peano. It no longer mattered what we thought. These so-called experts could not agree amongst themselves. What they did agree on was that it was a unique piece of work that had given them all food for thought.

When we got home, with the magic of the now rapidly expanding Internet, it didn’t take long to discover the truth. They were all correct! What we had represented was a Hilbert Open Peano Curve. We also found that there was a Sierpinski Closed Peano Curve so, naturally, we made that too.

It seemed sensible to drop all mathematical references from their titles. The first became Spacecraft, the other became Square Snowflake.

The idea of space-filling curves was fascinating and was to resurface later. It also took me back to my sixth-form days in school. The school was only a few years old when I first went there and most rooms had traditional wooden desks and chairs. However, the sixth form block was built after the rest and the rooms were equipped with the latest innovation – plastic laminate tables. These tables had a very pale squiggly pattern on them and I delighted in going over the pattern with a pencil. It was amazing how long it took to cover a very small area as the curves twisted backwards and forwards and were very tightly packed.  I hadn’t heard of space-filling curves then but I’m sure their complexity must have infiltrated my brain.