Bridges London: Mathematics, Music, Art, Architecture, Culture (2006)

Pages 521–524

Celtic knotwork is a form of decoration in use for over a thousand years. The designs fill spaces or borders with a pattern derived from plaiting. The designs have no loose ends and may contain more than one closed loop. As in a plait (or braid) of hair, each strand bounces back and forth like a billiard ball to form a pattern of diagonal lines between the edges of the rectangle while crossing over and under others alternately. The dimensions of a rectangular plaited panel can be expressed as the number of bounces there are along the long and short edges. The number of closed loops, referred to as knots by knot theorists, is the greatest common divisor of these two numbers. This paper shows how one can predict the number of loops there will be as a piece of knotwork is created from the panel by removing some of the crossing places and rejoining the loose ends, without crossing to make a gap either looking like ) ( to make what will be called a horizontal gap, or like this shape turned through 90° to make a vertical gap. As each of the chosen crossing places is removed and the loose ends rejoined in this way a more intricate interlaced design is formed. It is easy enough to trace round the resulting design with coloured pens to find out how many closed loops there are, but the results proved and demonstrated in the paper enable one to predict how the number of loops will change at each stage of the creation of the interlaced design. Such a prediction is not addressed by current knot theory. The first thing to notice is whether a loop crosses itself at the crossing to be removed or whether two different loops cross there. In the first case one or two questions must be answered before the number of loops can be predicted. In the second case the two different loops get combined into one single loop by the rejoining of the loose ends. The difficult part of the research was to devise questions which could be proved to make reliable predictions possible. One’s common understanding of how one might take a shortcut back to the start while following a nature trail provides the last link in the chain leading to the prediction. The method will be applied to successive designs produced as each crossing is removed.

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