Within textile practices, weaving is a relatively technical discipline. Because it works with fixed structures and patterns, mathematical principles provide a rich source of inspiration for new designs. For example, the Fibonacci sequence is often used to make color distributions in tea towels. But there is much more! This article explains how percolation on a hexagonal lattice can be translated into a woven pattern.
Weaving is one of the techniques used to turn loose threads into fabric. In weaving, you have a system of parallel lengthwise threads (the warp), which are bound by crosswise threads (the weft). The points where the warp threads cross the weft threads are called binding points. There are countless ways in which these binding points can be distributed, and that distribution largely determines the properties and appearance of the fabric.
Tablet weaving: a specialized weaving technique
A specialization in weaving is tablet weaving, a technique mainly used for making narrow bands. In this method, threads in four different colors are threaded through the four holes of a square card (tablet). By turning the tablet a quarter turn forward or backward, the color visible on the top side of the fabric changes. The tablet can also remain in the same position.
So, for the middle card in the drawing below, you have three possibilities:
You leave the tablet as it is; the red thread remains on top,
You turn the tablet a quarter turn forward; the light blue thread comes to the top,
You turn the tablet a quarter turn backward; the yellow thread comes to the top.
The thread in the top hole is visible on the top side of the fabric. The thread in the bottom hole appears on the underside. The other two threads lie in the middle layer of the fabric and are not visible on the outside.
Two weft threads are used to bind the warp threads: one in the top layer and one in the bottom layer. When the desired color lies on top, a weft thread is inserted in both layers. If a tablet is turned, the warp threads are bound; if the tablet remains still, no binding takes place.
The lines are the warp threads, the dots are the weft threads. Where the warp threads cross, there is a binding point.
To weave a band, multiple tablets are used side by side. Each tablet can be turned individually, as long as the rules described above are followed. This makes it possible to create very complex patterns, in which the four colors can alternate in many different ways.
Pebble weaving and percolation on the hexagonal lattice
A well-known structure in tablet weaving is the pebble structure. In this technique, the warp threads are bound in pairs after every three weft insertions. In the ground weave, this creates a pattern of characteristic dots, the so-called pebbles. By connecting these pebbles horizontally or diagonally, patterns can be formed.
The inspiration for the design described here came from the cover of the book Probability by Geoffrey Grimmett and Dominic Welsh, which Remco van der Hofstad uses for his lectures. As soon as I saw the cover, I knew immediately that I had to do something with it, and I borrowed the book for a few days so that I could work on a design.
I based the final design entirely on the principles of percolation on a hexagonal lattice. In the pebble structure, such a hexagonal lattice is implicitly present. In the picture below (left) you see a graph of the fabric in a pebble structure using 48 tablets and 21 turns. The weft is not depicted, since it is completely covered by the warp threads. In the middle the hexagonal lattice is projected on it. The faces of the hexagons can be colored using two opposite colors from the tablets; in this design, those are red and dark blue. The outlines of the hexagons are woven in yellow and light blue, colors that are also used for the pebbles themselves.
The pebble structure (left), the hexagonal lattice (middle) and the final pattern (right).
Next, an irregular and seemingly random pattern can be created by determining independently for each hexagon, in accordance with percolation theory, whether it will be red or dark blue. I did this using a die: an odd roll meant red, an even roll dark blue.
At the places where two adjacent hexagons differ in color, a phase transition arises. This transition can be represented by a yellow or light blue line. It was precisely at this point that I let go of the mathematical method; after all, I had four colors at my disposal. So the choice between yellow and light blue was not made according to a formal rule, but intuitively, based on the visual balance and rhythm of the pattern within the band as a whole.
And that is how an abstract mathematical model is transformed, thread by thread, into a tangible pattern.
Marieke Kranenburg (TU Eindhoven) has been project manager of the Gravitation Program NETWORKS since 2013. In addition to her work at the university, she is a professional weaver. In her designs, she enjoys drawing inspiration from the ideas, structures, and concepts contributed by her mathematical colleagues.
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