three directional weaving

Three Directional WeavingAuthor(s): William GibbsSource: Mathematics in School, Vol. 21, No. 2 (Mar., 1992), pp. 2-5Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214847 .

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Three Urectional

IJ'eavin by William Gibbs Overseas Education Unit, School of Education, University of Leeds

Fig. 2

A series of articles on the theme of paper patterns was published in this journal between May 1990 and January 1991. In the final article I touched on the idea of weaving in three directions and now provide extensions. The simplest and most common form of weave in which three strands are woven at an angle of 120 degrees to each other is the "hexagonal weave". This is used throughout the world; in the Seychelles to create fish traps (fig 1), in Bhutan to make carriers for vegetables (fig 2).

Fig. 1

This pattern is simple and elegant but it takes thought and concentration to weave. It starts from the basic node at which all strands in all three directions meet (fig 3);

Fig. 3

The pattern created is a mixture of hexagons and parallelograms and this becomes more obvious if three different colours are used in the weaving (fig 4). A different combi- nation of shapes emerges if thin coloured paper is used in

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Fig. 4

the weaving and the result held up to the light or fixed against a window (fig 5). The holes remain hexagons but now the surrounding shapes are equilateral triangles and rhombi.

Fig. 5

Fig. 6

A second three directional weave is used in basket making and the diagonal strands weave over and under the vertical at an angle of 120 degrees (fig 6). Here the three strands do not follow the same weaving pattern and a mixed tessellation of irregular hexagons and parallelograms is

created. From this weave it is possible to create emergent patterns of tessellating shapes by choosing to weave different colours in a particular order. For example if 4 different colours (a, b, c, d) are used in the vertical strips and alternating colours a and c on one diagonal and b and d on the other then a new pattern of polygons emerges (fig 7).

Fig. 7

Once again it is interesting to plait this weave using very thin paper of the same colour. It is possible to predict the pattern of stars and rhombi that will emerge but the result when held to the light is nevertheless surprising in its clarity (fig 8).

Fig. 8

At this stage one can consider whether or not the pattern is altered if strips of different widths are used. For example if the ratio of the widths of the diagonal strips to the vertical strips is altered from 1 : 1 to 1 :72 then the pattern becomes transformed to one of congruent parallelograms (fig 9).

A third example of a three way weave is even harder to plait (fig 10). Called "Anyam gila" or the "mad weave" (Harvey, 1975), this weave was given as a punishment to prisoners in Singapore jails so difficult was it thought to be. Here in these examples from Malaysia (fig 11) and

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Fig. 9

Fig. 10

Fig. 11

Fig. 12

Thailand (fig 12) the beauty of the pattern can be seen, the rhombi combining to give hexagons and creating an illusion of three dimensional cubes.

This particular pattern lends itself to an investigation of emergent tessellations and patterns that can be created from the basic weave by varying the number and order of coloured strands in each direction. The woven hat from Malaysia (fig 13) has been created from the "mad" weave by using strands of alternate colour in each of the three directions.

Fig. 13

This emerging tessellation is so delightful that it immedi- ately suggests an investigation. Can other tessellations emerge if different arrangements of colour and order are used and, if so, have these been discovered and woven by people in Malaysia or elsewhere? Here for example is a possible pattern, a mixed tiling of a six pointed star and rhombus created by weaving strands of two colours, a and b, in the order abaabaaba in each of the three directions (fig 14).

Here (fig 15) is another example, this time a mixed tessellation in which strands of just two colours are woven; aab in two directions and abb in the third.

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Fig. 14

Fig. 15

A fourth method of weaving in three directions is used by the Karamakoto Indians of North America and the pattern they create is a mixture of isosceles trapezium, equilateral triangles and rhombi (fig 16).

Fig. 16

Again by varying the order of colours woven in each direction new tessellations and patterns emerge. If three colours are used one after another in each of the three directions then three different tessellations can emerge

depending on how the same colours are combined. One is a tessellation of hexagonal arrows, another a tessellation of parallelograms and the third of hexagons (figs 17, 18, 19).

Fig. 17

Fig. 18

Fig. 19

The study of traditional and contemporary weaving shows how rich this field is and how susceptible to mathematical description and to mathematical investigation. Just as the simple two directional weave so commonly found throughout the world proves a model and example of a rectilinear grid and cartesian space so the three way weaves can be seen as way into the study of tessellation and pattern based on an isometric ordering of space.

References Collingwood, P. (1987) Textile and Weaving Structures, Batsford. Gerdes, P. (1988) On Culture, Geometrical Thinking and Mathematics

Education. Educational studies in Mathematics, 19, 2. Gibbs, W. (1990) Paper Patterns -4, Paper Weaving. Mathematics in

School, 20, 1. Harvey, V. I. (1975) The Techniques of Basketry, Batsford. Regensteiner, E. (1986) The Art of Weaving, Schiffer. Rosbach, E. (1986) The Nature of Basketry, Schiffer.

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three directional weaving

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