Mathematics of crochet and knitting

Summary: Crochet and knitting can be used to create models of mathematical surfaces.

Crochet and knitting are techniques for turning one-dimensional yarn or thread into two-dimensional fabric by knotting it in a regular pattern. Both produce flexible, elastic fabric, although crochet is firmer than knitting. Historically, crochet and knitting were used to produce both functional and ornamental textiles by hand, but both are now hobby pursuits.

Since both techniques produce regular arrays of stitches, they can be used to display a wide variety of symmetric patterns. Furthermore, both can be used to make intrinsically curved fabrics. This allows mathematicians and others to approximate or replicate the geometry of hard-to-visualize objects, including models of two-dimensional mathematical curved surfaces, such as spheres, tori, or sections of the hyperbolic plane. Crocheting and knitting circles have been held at professional mathematics conferences for both recreation and serious discussion of mathematical concepts. Mathematician Carolyn Yackel has noted, “Knitting and crocheting are helping us think about math we already know in a different light.”

Crochet

In crochet, stitches are made by pulling loops of yarn through each other with a hook. One stitch is worked at a time. Every crochet stitch is attached at its base to an earlier stitch. Varying the type of stitch and the way new stitches are worked into earlier stitches can produce many different patterns. Crochet can be worked back and forth in rows or in circular rounds. Working two stitches into one base stitch increases the number of stitches and makes the fabric wider; decreasing the number of stitches reduces the width of the fabric. Placing increases or decreases at the edges of the work makes flat fabric with curved edges. Placing increases or decreases in the middle of the fabric makes it intrinsically curved.

The origins of crochet are not well understood. Few—if any—samples are known from before the nineteenth century. At that time, it was generally worked in fine cotton or linen thread and used for lace edgings, doilies, and other household textiles. From the middle of the twentieth century on, crochet has generally been worked in thicker yarn. It is often used to make blankets known as “afghans.” The hobby of crocheting stuffed animals, known as “amigurumi,” has spread around the world in recent years; because of the curved shape that these toys are crocheted in, they have few seams.

Several mathematicians have designed crocheted models of mathematical curved surfaces. As mathematician Daina Taimina has pointed out, it is especially simple to crochet negatively curved surfaces, such as a hyperbolic plane; the crocheter simply works an increase (an extra stitch) once every two or three (or n) stitches in every row. These increases cause the fabric to fold back on itself rather than lie flat. The closer together the increases are, the more ruffled the fabric.

The Hyperbolic Crochet Coral Reef, a project by the Institute for Figuring in Los Angeles, is intended to increase awareness of global warming issues by bringing together mathematicians, marine biologists, and community crafters in a highly visible way. The project asks volunteers to crochet models of coral reef life forms using Taimina’s patterns. This effort and other mathematical crochet or knitting projects have been used successfully by mathematics educators in their classrooms.

Knitting

In knitting, as in crochet, stitches are made by pulling loops through each other. Knitting can also be worked in either rows or rounds. Two (or more) needles are used and many stitches are held on the needles simultaneously. The most basic stitches are “knit” and “purl” and there are techniques for increasing, decreasing, and making textural elements such as holes, cables, or bobbles. Knitting produces a flatter, stretchier fabric than crochet. (Indeed, most elastic fabric produced today is machine knitted.) As with crochet, increases and decreases allow the knitter to change the shape and curvature of the fabric. The shaping and elasticity make knitting ideal for garments such as socks, hats, gloves, and sweaters where both fit and comfort are important.

Hand knitting was once an important industry in Europe. Medieval guilds produced stunning garments for the wealthy in the Middle Ages, and a large cottage industry knitted stockings in the eighteenth and nineteenth centuries. Written patterns become available in the nineteenth century, and ornate knitting in fine thread became a popular pastime for ladies.

Hand knitting resurged in popularity in the first decade of the twenty-first century. Many current designers of garments and home textiles take their inspiration from mathematics, using symmetry and geometry to create attractive garments and household items.

Like crochet, knitting can be used to produce curved mathematical surfaces. Wide, soft, knitted Mobius bands are often knitted for use as scarves.

Bibliography

Belcastro, Sarah-Marie, and Carolyn Yackel, eds. Making Mathematics With Needlework. Wellesley, MA: A K Peters, 2008.

Bordhi, Cat. A Treasury of Magical Knitting. Friday Harbor, WA: Passing Paws, 2004.

Gaughan, Norah. Knitting Nature: 39 Designs Inspired by Patterns in Nature. New York: STC Craft/Melanie Falick Books, 2006.

Obaachan, Annie. Amigurumi Animals: 15 Patterns and Dozens of Techniques for Creating Cute Crochet Creatures. New York: St. Martin’s Press, 2008.

Osinga, Hinke, and Bernd Krauskopf. “Crocheting the Lorenz Manifold.” Mathematical Intelligencer 26, no. 4 (2004).

Taimina, Daina. Crocheting Adventures With Hyperbolic Planes. Wellesley, MA: A K Peters. 2009.