Textiles and mathematics

Summary: Mathematics is integral to creating both traditional and modern weave patterns in textiles.

Textiles are flexible sheets made out of fibers. Natural textiles are made from plant fibers, animals, or minerals; artificial textiles use human-made fibers, like plastic or synthetic proteins. Woven textiles combine longer fiber threads either by hand or by using looms or knitting machines. In nonwoven textiles, like felt, short or microscopic fibers are bonded by chemical or physical treatments. Nonwovens are often meant to be highly durable or disposable and have many applications in health, construction, and filtration technologies. Mathematical methods are used to design, produce, and analyze textiles. In 1804, Joseph Jacquard invented a weaving system using cards with patterns of holes to control loom threads. These cards were later modified by Charles Babbage into computer punch cards. Weaver and mathematics teacher Ada Dietz wrote Algebraic Expressions in Handwoven Textiles in 1949. She outlined a method for using expansions of multivariate polynomials to generate weaving patterns.

Weave Formulas

On a loom, “warp” threads are held parallel and “weft” threads are passed over and under them. A pattern formed in one pass of weft can be either repeated exactly, transposed, or otherwise changed in the next passes. Let A stand for warp threads on top and E stand for the weft thread on top. In plain fabric, a pattern AEAE… indicates that the weave is transposed by one thread in the next row. Basket weave uses AAEEAAEE… , so the pattern is repeated for two rows and then transposed by two (or some other whole number) for the next two rows. Satin is AAAAEAAAAE… , giving four repeats followed by one transposition. A satin weave results in the majority of the threads being parallel, so light is minimally scattered, producing the characteristic sheen. In contrast, twill has a distinct, textured diagonal pattern formed by using an EEAEEA… weaving scheme. Patterns may be added to plain weaves by printing or dying the fabric. The U.S. group Complex Weavers provides a forum for sharing advanced weaving methods and patterns, such as manifold twill.

Patterns and other factors like the thread intersections per area also dictate other properties. For example, plain weave fabrics tear the easiest, because force is applied to the single thread immediately next to the tear. Crimp is how easily the fabric morphs under tension. Plain weaves generally morph the easiest. Wrinkle resistance is the opposite; the more freedom of movement threads have, the easier it is for them to return to smoothness. Satin is an example of a wrinkle-resistant weave. On the other hand, satin silks shrink the most because their weave pattern is loose. Twill has a relatively high resistance to tearing, which makes fabrics such as jean popular for working clothing.

Cultural Textiles

Textiles are a significant cultural art form for many people in Africa. The three most well-known forms are kente, adire, and adinkra. Kente cloth is woven in long narrow strips, traditionally by Asante and Ewe men, and then sewn together into larger pieces of fabric that may be used for clothing or household goods. The cloth was often a sign of wealth and kept for special occasions. There are more than 300 known kente patterns, many of which represent people or historical events. Widely found adire cloth has patterns made by resistance dying. The cloth is tied, stitched, or stenciled, often with geometric patterns, to prevent the dye from adhering to some portions of the cloth. Adinkra cloth is printed, usually by drawing a square grid and stamping symbols into each square. This highly developed symbol language expresses concrete and abstract concepts, such as transformation or unity. Like kente cloth, adinkra often tells stories or proverbs. Tessellations and other repeating patterns are also common. In Ghana, the cloth was originally worn for mourning and some is still reserved for that purpose.

In Scotland, tartans represent families, clans, or regions. A “sett” is a specific plaid pattern, specified by sequences and widths of colored stripes. The pattern is formed by interweaving bands of stripes at right angles. Most are symmetrical, which means the sett is reflected 90 degrees around a pivot or center stripe. Asymmetrical setts have no pivot point. Symmetry has implications in kilt making. A kilt “pleated to the sett” has pleats folded to visually reproduce the tartan pattern across the back of the kilt, often not possible with an asymmetric pattern. Tartan patterns have been investigated with mathematical methods, such as group theory, and they are used in classrooms as examples of symmetry. Artist Andrew Hennessey has proposed “stella tartan” in which tartan setts would be woven radially and overlap in irregular polygon patterns.

High-Technology Textiles

The Industrial Revolution made rapid mass production of textiles feasible and the textile industry has since used many mathematical and computational techniques to continue its evolution. These techniques include differential equations, numerical methods, image processing, pattern recognition, and statistics. Computer-aided design (CAD) and computer-aided looms (CAL) are widespread. Application areas include supply chain management, quality control, and product development. The latter may involve structural modeling and simulation, as well as thermal or biomechanical bioengineering, particularly for specialty textiles. Some competitive swimwear has tiny triangular projections that mimick shark skin to reduce drag. An absorbent, nonwoven textile called air-laid paper is used in diapers. Integrating tiny light-emitting diodes into fabric allows clothes to change color or display text or animation. Thermal self-regulation may be achieved with phase-changing microcapsules that become fluid for cooling or solid to release heat, as needed. Weak link theory and bundle theory, as well as research in twisted continuous filaments, helical modeling of yarns, two-dimensional elasticity theory, aerodynamics, and many other investigations have also revolutionized the individual threads that compose fabric, often changing its properties even when using traditional weaves.

Bibliography

Dietz, Ada. Algebraic Expressions in Handwoven Textiles. Louisville, KY: The Little Loomhouse, 1949. http://www.cs.arizona.edu/patterns/weaving/monographs/dak‗alge.pdf.

Harris, Mary. Common Thread: Women, Mathematics and Work. Staffordshire, England: Trentham Books Limited, 1997.

Zeng, Xianyi, Yi Li, Da Ruan, and Ludovic Koehl. Computational Textile. Berlin: Springer, 2007.