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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

The faithful Curious and Interesting Geometry next revealed another possibility that sent us in several directions at once. ‘Hinged tessellations’ are shapes that will fit together to cover the surface but they are only joined at some of their corners so they can swing apart to make interesting patterns of shapes and holes. We were only interested in the squares. Pairs of squares are hinged alternately at top or bottom so that the whole pattern can be pulled apart to leave rhombus-shaped holes between. To begin with the rhombuses are very skinny and, as the squares are pulled further apart the holes become squares. It was the movement of this structure that fascinated us and, in the first instance, we had no thought of representing it in our usual way.

We had grand plans to make an artwork that would move through the various stages simply by pulling a cord. We scoured haberdashery, gardening and diy shops to find suitable mechanisms. The squares were easy. They were clear plastic with sparkly hologram foil on one side. They had holes in the corners and split rings, normally used for jewellery, in the corners to join them together. The whole assembly was attached to a pole with cords passing through at strategic points. It worked in theory but we were never happy with the practical operation and it was eventually abandoned.

However this proved to be  another opening onto a new path and ‘things that move’ took over our thoughts. It wasn’t possible to recreate the movement of hinged squares, in yarn. We could cope with the fully open or fully closed positions and, by adding beads to every square we could make the same hanging do both of these, but nothing in between. The hanging has 64 squares in one colour and 81 in another colour and we were quite surprised to realise that we needed to use two square numbers to make the larger square pattern. The beads act as buttons, the squares have holes that act as buttonholes. In the opened-up state the hanging is a pattern of squares with square holes in between. When closed the squares stand on their points with no spaces. It takes a very long time to fasten all the beads! As a work of art, rather than an item to keep you warm, it looks particularly intriguing if left with some of the bottom squares unfastened, as though they are about to fall off.

We had often made paper hexaflexagons in school. These also come into the category of ‘things that move’. A long strip of paper is folded to create a line of nine equilateral triangles. There is a nice mathematical trick that makes it much easier than it sounds so it is always a fun thing to do. The ends of the strip are joined, in a particular way, to form a hexagon.

The hexagon can then be twisted and turned to reveal some magical properties. It is also a good vehicle for teaching about angles and shapes. There are more complex variations of the strip and the extent to which this can be developed depends on the age and ability of the class.

Making simple hexaflexagons, in paper, with children, always has the same results. A couple of children will make the flexagon immediately. These are very often pupils who struggle with other aspects of Maths. There are always a few who get frustrated at the folding and sticking stages but those who have already managed it are quick to offer assistance.

The next tricky bit comes when they try to flex the flexagon. It needs to be thoroughly folded, in both directions, along all the folds. It has to be soft and supple before it will flex and many try to rush this stage. Once it is supple, it can be opened and it appears to turn through itself and flatten out again. This might seem to be the end of the exercise but there is more to come.

Looking at the hexagon it can be clearly seen that there are six equilateral triangles on each side. On one side, mark the points at the centre. On the other side mark the centres with a different colour. Flex the hexagon and the marks move, as if by magic. Some move to the outer points, others disappear completely. Every time a new centre is revealed it is marked with a new colour. For most, it is a matter of luck whether they find the same centres time and again or manage to find all the different possible centres.

We had been having fun with these for years and only now thought of adding them to our growing collection of knitted and crochet items. It took quite a while to work out a way of doing it. Although there are only nine triangles in a paper hexaflexagon, there are two sides to each so these had to be regarded as 18 triangles. It was extremely difficult to work out where the colours would lie on the original straight strip. The easiest way to do it was to cheat. We made the paper version, added the colours and cut it apart again to lay flat as a strip. Using this as the pattern for the colours, each triangle was made from a crochet hexagon with triangles, in different colours on three of the sides. The fronts and backs of the resulting triangles were joined, according to the pattern, with wadding sandwiched between. This flexagon became a cushion. It isn’t quite as easy to flex as the paper versions because it is so soft but it is easily pushed back into shape. I have taken the cushion into many classrooms (and groups of adults) and, again, the reaction is now totally predictable.

The first person to pick it up always puts it on their head. They immediately realise that there is a gap in the middle and it can be pulled over the head to form a collar. Eventually it gets used for its proper purpose but all this adds to the fun and allows ideas to be assimilated without pressure.

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23a. THINGS THAT MOVE