Quilting and mathematics

Summary: Quilting can incorporate and help teach mathematical concepts, such as symmetry and tessellations.

Quilting is a needlework technique in which two layers of fabric are sewn together, usually with an inner layer of padding (called “batting”) between them. Often, one or both outer layers are formed by sewing together (or “piecing”) smaller pieces of fabric. Sometimes, designs are appliquéd (sewn onto a larger piece of fabric) or embroidered on the quilt. The quilting itself (the stitches holding the layers together) is often also decorative. Many traditional quilt designs display mathematical concepts, such as symmetry and tessellations, that generalize into the abstract mathematics of group theory and tiling theory. In diverse parts of the world, people create quilts not only to warm the body at night, but also to use as clothing, furnishings, or to share family or cultural history. A carving of an ancient Egyptian Pharaoh figure containing what may be a quilt and a quilted carpet found in the mountains of Mongolia dates to approximately the first century. Directions can be found to quilt coded designs that may have been used on the Underground Railroad.

Quilt Designs

Some traditional quilts are “crazy quilts” in which scraps of fabric are sewn together in no particular pattern. Others are formed of similar or identical square “blocks,” each of which may be pieced together. Often, quilt patterns involve careful measurement (using common fractions) in the cutting and sewing of the pieces.

Quilt designs are often symmetrical—the entire design can be folded in half along a line such that one half falls directly onto the other half. Each half is a reflection of the other along that line, which is called a “line of symmetry.” These lines may be vertical, horizontal, or diagonal. Some quilt blocks, such as the traditional Amish Star, are symmetric along many lines. Quilts and quilt blocks may also have rotational symmetry—the design can be rotated around a point through less than a full rotation in a way that leaves the overall design unchanged. Quilts in the Hawaiian Islands are known for their distinctive radial symmetry.

Mathematics generalizes this everyday concept of symmetry. A mathematical object (not necessarily a geometric shape) is symmetric with respect to a particular mathematical operation if the operation, applied to the object, preserves some property of the object. A mathematical group consists of a set of operations that preserve a given property of a given object. Group theory is central to abstract algebra and has many applications.

Fabric quilts, construction paper versions, or computerized models of quilt designs have been used to introduce students as early as elementary school to geometric concepts, such as symmetry and transformations. They help children develop, at a basic level, fundamental algebraic properties, such as inverse, identity, and equivalence. Students also make quilts to explore many other concepts, such as the Pythagorean theorem, polar coordinates, group theory, the Fibonacci sequence, and Pascal’s triangle, named after mathematician Blaise Pascal.

Tessellations

A tessellation (or tiling) is an infinitely repeating pattern composed of polygons covering a plane without any openings or overlaps. Many quilt designs are formed from tessellations. A regular tessellation uses one polygon with equal sides and equal angles, such as equilateral triangles, squares, or regular hexagons. For example, the traditional Grandmother’s Flower Garden and Honeycomb quilt designs use tessellations of regular hexagons. Many modern watercolor quilts use tessellations of one-inch squares.

A semi-regular tessellation uses a combination of squares, triangles, and hexagons that are arranged identically around each vertex. Demi-regular tessellations, with two vertices in each repetition, form more complicated quilt patterns. Many quilt blocks, such as Log Cabin variations, consist of non-regular tessellations.

Mathematicians have generalized tiling theory to higher dimensional Euclidean spaces and to non-Euclidean geometries. These generalizations reveal links to group theory and to classical problems in number theory. Much of the art of M. C. Escher is based on non-Euclidean tessellations.

Other Designs

Contemporary quilters like mathematician Irena Swanson have also incorporated other mathematical concepts in their designs, such as infinite geometric series and fractals, as well as portraits of mathematicians. Mathematician Gwen Fischer created quaternionic quilts to visually showcase the algebraic structure of the group. For example, the lack of reflection symmetry across the main diagonal highlights the lack of commutativity of the group elements.

Bibliography

Fisher, Gwen. “Quaternions Quilt.” FOCUS 25, no. 1 (2005).

Meel, David, and Deborah Youse. “No-Sew Mathematical Quilts: Needling Students to Explore Higher Mathematics.” Visual Mathematics 10, no. 2 (2008).

Paznokas, Lynda. “Teaching Mathematics Through Cultural Quilting.” Teaching Children Mathematics 9 (2003).

Rosa, Milton, and Daniel Orey. “Symmetrical Freedom Quilts: The Ethnomathematics of Ways of Communication, Liberation, and Art.” Revista Latino Americana de Etnomatemática 2, no. 2 (2009).

Venters, Diana, and Elain Ellison. Mathematical Quilts—No Sewing Required. Emeryville, CA: Key Curriculum Press, 1999.