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The real new millennium began. This was 2001, not 2000 and, with three mathematicians now in the house we would have argued this point with anyone.  Ben came to stay for a few months. He isn’t really a mathematician. He is a physicist, putting maths, and other things, to a practical use. He had always been involved in the design and creation of the afghans and was instrumental in informing the world of what we were doing, via the web site. Little did he know, when he came to stay, that our afghan designs would play a part in where his path through working life might begin.

For the Chinese, 2000 had been the Year of the Dragon. For us 2001 was dragon year. Early in the year, I had to invigilate in a school exam. According to the regulations, exam invigilators should not do anything other than give their entire attention to the students they are supervising. This was an exam with six or seven candidates who were spread around a classroom full of desks. I was there for two hours. There was no room to walk about without disturbing them and nothing to relieve the boredom. An invigilator is in grave danger of falling asleep in such circumstances so I took my trusty Curious and Interesting Geometry with me. I had looked at the pictures in this book so many times before it is difficult to believe that I could find something new.  I had recently been crocheting, rather than knitting, and perhaps I was looking from a different perspective. Dragon Curves entered my consciousness and I couldn’t wait to get home to look at them in more detail. When I did we were all hooked and an addiction began.

The concept is so simple but the potential is amazing. Take a long strip of paper and fold it in half. You get a strip of paper with a fold in it! Refold the paper then fold it in half again. When you look inside you have two folds going in one direction and one fold going the other way. Refold and fold in half again and, when you unfold, you get even more folds, some going one way, some the other. Keep a record of where the folds occur after each folding and it becomes obvious that a pattern is being formed. It becomes possible to predict what will happen with the next folding. In real life, the strip of paper can only be folded about six times before it gets too thick to bend. Mathematically, you can go on folding the strip as many times as you want.

These folds are interesting in themselves but the patterns the strip creates are even more intriguing. Unfold the strip and make a firm crease along each of the fold lines, taking care to maintain the correct direction for the fold. Stand the strip on its edge, with each fold  forming a 90 degree angle, and look down on it. You will see an intricate pattern of squares, and twists and turns. The strips get the name of Dragon Curves because, after a few folds, they start to take on the appearance of a dragon with large head, legs and tail.

These are space-filling curves in much the same way that Peano curves (as in Spacecraft and Square Snowflake) are space-filling curves. Each time the strip is folded the sections between the folds get shorter and the pattern becomes more intricate. The sections can be packed closer and closer until, theoretically, they cover the surface. Delving deeper revealed that two dragons joined head-to-tail produced the next size dragon. Joining four dragons head-to-head produces a beautiful four-armed ‘spiral’. Dragons interlock to cover the surface, when they are placed tail-to-tail. Four dragons placed head-to-tail make a closed ring with space for smaller dragons inside. The design possibilities were endless.

Steve was fascinated and spent his time drawing dragons, using various computer programs to make the task a little easier. Ben ventured into the deeper realms of changing the angles of the folds in the strip. This is almost impossible in a real-life situation, with strips of paper, but this was another of those occasions where maths has the advantage. In minutes he wrote a computer program to generate all the possibilities he envisaged. I couldn’t wait to get started on turning these beautiful curves into something that lay somewhere between what could be done with paper and what could be done with drawing.

Another aspect of mathematics came into play. According to the Four-Colour Map Theorem, it is possible to colour any design using no more than four colours, although you may choose to use more for some projects. When dragons of the same size are placed together, four of them meet at a point. If one is larger than the others there will only be three colours meeting but four colours may still be needed elsewhere in the design. Any afghan needed at least four colours although I wanted more as I wanted to represent each size of dragon as being different from the rest. Before decisions had to be made about the colours, we needed a background grid on which to work the dragons. While I set about making a filet crochet grid, Steve and Ben worked on a pleasing design that would fit into a square

The grid was tedious to make but the prospect of adding the dragons was a spur to finish it quite quickly. The yarns were chosen. It was to have four basic shades of mohair – orange, bright pink, vivid turquoise-green and purple. We decided to use mohair as it would fluff up to fill the spaces without showing any of the background. With the four colours we would be using another mohair yarn for each different size of dragon. All ‘order one’ dragons had pale green mixed with them, all ‘order two’ dragons had grey, and so on.

Completing the first dragon was a nightmare. The background had to be turned for every section of dragon, which was 512 times for the largest! Once the first one was in place it got much easier as mistakes were obvious when dragons didn’t fit snugly together and any errors were easily pulled undone. It wasn’t long before everyone wanted to have a go at adding the stitches. The process was so easy anyone could do it after one minute’s tuition. It was finished in no time and was the most spectacular thing we had ever created. We could have stopped there but the obsession had taken over and we went on, searching for anything with a square grid that could have dragons applied to it.

There was another afghan using the same design as the first but in just four colours – red, black, silver grey and dark grey – with no regard for the different orders of dragons. We made scarves and rug and bags and anything else we could think off, with various combinations of dragons. I have always wanted to add dragons to someone’s chain link garden fence, using strips of carrier bags, but I haven’t found a suitable place – yet. The afghans went into a booklet entitled Scaled Up, the smaller items became Scaled Down. Mainly for our interest, and to remember what we had discovered, we wrote another booklet, Dragonometry, which was purely about the mathematics involved. Much to our surprise we sell as many copies of that booklet as the others, even to knitting and craft groups.

We really were addicted to dragons but you may be wondering why this was such a significant time for Ben. Firstly, the booklet of instructions for these afghans was a little different from the others. It said ‘Designed by Steve Plummer & Ben Ashforth. Created by Pat Ashforth’.  It was the first time Ben had his name on a booklet. More importantly, he went for an interview as a Research Scientist, at a highly-scientific establishment. He had to give a presentation on a subject of his own choice. To my horror he chose to talk about crochet dragon curves – and got the job. The unconventional nature of the presentation, and the fact that he had learned two computer languages to compile the web site, was exactly the kind of eccentricity they were looking for.

SCALED UP