Bedspread mathematics

Today was a good day: the Ladybug, the Queen B, and I went to the Black Hills Quilters Guild Quilt Show.

Yes, I went to a quilt show and liked it.

“Quilts?” you say. “Those hideous patchwork blankets made by your grandmother that reek of mothballs and small dogs?”

Yes, those things.

I’ll have to admit, last year when the Queen B asked if I wanted to go to a Quilt Show, my immediate thought was that it sounded almost (but not quite) as much fun as stapling my gonads to the wheels of a Mack truck and letting it drive. On my mental list of “things to do,” wandering through acre after acre of ugly blankets is right next to experiencing tooth extraction using rubber bands and a corkscrew. Nevertheless, the Queen B had taken an interest in quilting during the seemingly endless adoption wait, and being a good husband I begrudgingly went along.

But when I first walked into the convention hall, I saw… Well, a damn ugly quilt, actually. Something like this:

But thankfully, next to it were a number of much more interesting quilts next to it, such as the Euclid-inspired “sun,” with its collection of circular tangents and chords,

or the quilted version of Q-bert’s gamespace

or this quilt which reminded me of Escher’s Whirlpools:

What struck me about these quilts was interesting mathematical shapes and images they contained. In fact, when I went to look at the original, boring quilt a second time, I discovered several delightfully unexpected details. First, take a closer look at that first quilt:

Note that the design of the whole quilt is merely a tiling of the plane by a single square design, sequentially rotated and/or reflected. Even more, the basic “atomic” square, called a log cabin in quilter’s terms, is actually a nice proof without words that the sum of consecutive odd numbers is always a perfect square; here is a proof with words demonstrating the general idea with applets.

Further inspection shows that the “top-stitching” — the threading used to connect the top patterns to the back of the quilt through the padding — takes the form of very basic polar rose curves of the from and $r=cos(2t)$.

So, whether or not she knew it, the quilter had infused her blanket with tidbits of considerable mathematical interest. It turns out that this is actually not uncommon in quilts: from the color patterns of the quilt blocks to the intricate curves in the top-stitching, quilts can communicate a great deal of mathematical insight*… and keep you warm, as well! I spent the rest of that day making interesting mental notes about the lovely intersection between the art of quilting and the art of mathematics. That carried over to today, and I figured it’d be neat to show you, valued reader, some of the gems I found over the past two years. Let’s go.

* In fact, this should have come as no surprise to me. Amy Szczepanski, a friend of mine from graduate school, once made a quilt that encoded pretty much everything about the non-abelian group of symmetries of the square (i.e. the dihedral group ). The quilt blocks themselves formed a visual representation of the 8×8 group multiplication table, whereas the top-stitching indicated the quotient group formed by mod-ing out the center. She showed if off at the Baltimore AMS-MAA Joint Mathematical Meetings; it was pretty sweet.

Tessellations

Perhaps not unexpectedly, quilts are very good at illustrating tessellations of the plane; after all, their most generically obvious attribute is that they’re formed by tiling together various squares of fabric. Many quilts at the shows featured very basic “pinwheel” tilings of the plane. Most of the pinwheel quilts I saw based their tessellations — again, not unexpectedly — on square pinwheels, ranging from the basic cross-like tilings of this quilt

to the slightly more rambunctious pinwheels of this one

to the positively ornate spiraled “snowflake” pinwheels of this one.

Other quilters broke free of squares and used hexagonal tilings, such as with the flowers of this quilt:

The quilt below tiles the plane using two different squares, a large one (with the ornate “flower” in the center and “starbursts” at the corners) and a small one. Such a tiling of the plane might seem pretty boring on first inspection, but it actually gives a nice geometric proof of the Pythagorean Theorem, if you know where to look.

As a final example, take this tiling of the plane by this very irregular curve through horizontal translations… and its periodic in color, to boot!

Granted, that’s probably pushing the limits of the tessellation idea, but if you want more mathematical connections from this quilt, I’d submit that it also does a fabulous job of visually depicting what a family of antiderivatives looks like, provided you turn your head sideways.

Curves

Speaking of graphs (albeit indirectly), nothing says “math” more than curves. Just as the quilt blocks naturally lend themselves to describing interesting plane regions, the quilt’s top-stitching naturally describes interesting curves in the plane. As illustrated in the “boring” quilt example above, polar “rose” curves, especially the four-petaled kind, were a common top-stiching design:

I was greatly amused by this quilter’s much more interesting take on a “periodic rose curve.” (Extra credit to whoever can find the parametric equations that generate it!)

Some quilters seem to delight in turning the whole of their top-stitching into devilishly complicated and wonderfully ornate Jordan curves, which reminded me of Robert Bosch’s** beautiful Travelling Salesman Problem Art:

** As an aside, I met Robert when I was teaching at Colorado College when he presented his complete 24-set domino portrait of emeritus professor Dave Roeder. If ever there was a guy who can create art beautiful to the masses from mathematics beautiful only to mathematicians, it’s Robert Boch.

I was much more surprised to see a curve depicted as the actual quilted image itself, rather than hidden away in the (frequently unnoticed) detail work of the top-stitching. For example, this quilt gives an evocatively atmospheric plot of the sine curve

whereas this quilt does a nice job depicting a saw-tooth curve, the kind students love to compute Fourier coefficients for. (It also has some beautiful, if not exactly mathematical, top-stitching in it.)

Fractals

Many quilts utilized some aspect of fractals or self-similarity in their design. For example, this quilt contains several fractal-like square spirals. Each spiral, if carried out to infinity, is composed of infinitely many scaled-down (but otherwise identical) copies of the original spiral. Constructions like these have wonderful connections with infinite geometric and p-series ideas in calculus.

Many quilts used the fractal idea very subtly, such as this quilt in which the whole of the orchid is composed of non-discrete sub-orchids, if you will:

Other quilts were damned explicit about their fractal structure:

Optical illusions

A final interesting theme in quilts I saw was presenting the illusion of depth. Some quilts did this by “quilting” classic optical illusions, such as this one using the “Missing corner cube” illusion:

This quilt gives an even better illusion of depth is presented by “stacking” these cubes (a la Q-bert again):

Some quilts tried to give the illusion of depth by including very crisp, well-defined images with equally crisp, well-defined shadows. A number of quilts used this technique to give the illusion of leaves falling “above” the plane of the quilt itself, some very effectively:

In contrast, some quilters opted to breathe depth into their work by using a diffused mosaic technique, depicting abstract undulations with a large pallet of colored rectangles. This quilt, for example, gives a convincing sense of depth among its twisting bands:

And while the quilter probably didn’t intend it, this quilt (when looked at at a distance) exhibits some peripheral drift illusion; in fact, it reminds me very much of Akiyoshi Kitaoka’s dizzying “Rotating Snakes”.

Conclusion

This is just a small sampling of the mathematical art I saw inside the quilts. I’d encourage you to did up Gramma’s old quilt and take another look at it to see what mathematical treasures are hidden away in it. Just don’t gag on that mothball smell. Yech.