Most of us see quilts as a way to stay warm or to brighten up a bed. Gerda de Vries sees math.

When we think of math, we might think of long equations that you need a pencil and calculator to solve. But de Vries, a UAlberta mathematics professor and self-taught quilter says that math can be as simple as the practice of geometry, which asks and solves questions about how shapes fit together.

Quilting, particularly without a pre-set pattern as de Vries often does, requires a mathematical mindset. To create a pattern, a quilter creates a set of rules, then determines how many ways the rules can be satisfied. For example, a quilter might set rules for themselves by seeing how many ways they can rotate one block (a square piece of fabric) without repeating the same rotation twice.

“Sometimes, you just make a bunch of blocks and play with them and the design comes out of that. That’s very mathematical,” she says.

De Vries says the patterns in quilts might appear to be random, but they’re actually quite calculated — even though they don’t need an actual calculator to make. By finding the patterns in these quilts, then, we are doing geometry without even realizing it.

Here, we present five math lessons with the help of de Vries’ mathematician’s eye and quilts from the Rosenberg Quilt Collection at UAlberta’s Department of Human Ecology.

### Lesson 1: Tessellation, Part I

A tessellation is any collection of two-dimensional shapes that fills a space with no overlaps or gaps. All quilts are essentially tessellations because they’re an assortment of fabric tiles that fit together like a puzzle. Even the crazy quilt below, which has no repeating pattern of shapes and colours, is a tessellation.

### Lesson 2: Tessellation, Part II

Patterned tessellations like the Drunkard’s Path below most fascinate de Vries. Notice how the shapes in red and white, despite their complex edges, fit neatly together in a uniform pattern across the entire surface.

### Lesson 3: Symmetry, Part I

Symmetry is another common mathematical function seen in quilts. With one simple block, a quilter can create myriad patterns based on how each block is positioned in relation to others.

In the first image below, each block with a purple square and blue parrallelogram is a cell (shaded area). The way the cells fit together creates a unit (orange outlines).

The same cell can create very different patterns. Math recognizes four “operators of symmetry,” or ways of arranging the cells (images below) — translations, reflections, rotations and glide reflections.

### Lesson 4: Symmetry, Part II

You can find two types of symmetry in the Martha’s Choice pattern, depending on how you frame the cells. Rotation symmetry creates the flower shape where four cells join. Reflection symmetry comes into play if you shift the frame so the points with the triangles come together, creating a white hourglass and a red hourglass shape that form a diamond.

### Lesson 5: Fractals

A fractal is a never-ending geometric pattern or object that repeats infinitely at any magnification, meaning it’s “self-similar.” As you zoom in closer, you find smaller and smaller copies of the whole.

This Nine Patch quilt could be seen as the beginning of a fractal, with the quilt’s squares divided into smaller squares an infinite number of times. The four squares are divided into beige, black and red squares. Although the differently coloured squares are different sizes, they all have the same proportions — each square’s sides are all the same length. This means they are self-similar.