### For the love of symmetry : Fiber Arts

Let me begin, somewhat circularly, by quoting myself:

“I like symmetry. Or, to be more precise, I like

symmetries.You see, I’m a mathematician, and mathematicians find patterns everywhere. We can’t help ourselves. We don’t just admire things that are symmetric, we ask questions about how they are symmetric. By a mirror reflection, like a face? By a rotation, like a pinwheel? By a translation (repetition in a straight line), like a row of toy soldiers? By a glide reflection, like footprints in the sand?”

This text is the beginning of a blatant exercise in smuggling higher mathematics into the world at large: the scarf pattern, *Crystalline*, that I contributed to the online knitting magazine *Knitty*. (The pattern is still available free of charge to any interested knitters.)* *In recent years, I have been increasingly absorbed by representing different symmetry structures in various fiber arts: knitting, embroidery, beadwork, and so forth. The underlying mathematics is fascinating and often interacts with each handcraft in subtle ways. Plus, it’s always nice when you can prove a mathematical proposition with a scarf.

## Generating symmetries

*Fundamental Frieze Scroll II*

For a clearer view of the various symmetries, let’s look at a more recent knitted artifact, the beaded lace wall hanging *Fundamental Frieze Scroll II*, pictured and diagrammed here.

I designed this to better explain symmetry patterns, using beaded points and lines to mark the symmetries in the seven lace designs. Each design has a translation symmetry, meaning that it repeats over and over again in a single direction, and the remaining symmetries are applied to the theoretical pattern that continues to repeat forever at both ends, running off of the scroll and into infinity. Mathematicians call such designs frieze patterns and catalog each possible symmetry structure as a frieze group.** **White lines mark reflection axes: if you mirror the pattern across a white line, the shape of the design remains unchanged. Yellow lines mark glide reflection axes: here, a mirror across the line shifts the design, but if you glide (translate) it slightly along the yellow axis, you can restore its original appearance. At each blue bead, a half-turn rotation centered at the bead preserves the design.

We have known for a long time that there are exactly seven different symmetry structures that a frieze pattern can have; these are the seven frieze groups. *Fundamental Frieze Scroll II* is a complete symmetry sampler, containing one design for each of the symmetry groups. Meanwhile, the *Crystalline* scarf exhibits some of the closely related wallpaper groups, which describe the symmetries of patterns that repeat in two independent directions, filling the entire plane just as someone with unlimited time might wallpaper an infinite room. There are seventeen wallpaper groups, but only nine of them fit into the non-square rectangular grid of double knitting, and these are the nine depicted in *Crystalline*.

*Fundamental Frieze Scroll II* uses a very particular method to construct the different symmetry types. In the scarf, I was a little more ad hoc, taking advantage of the natural symmetries of the hearts (reflections), vines (glide reflections), and scrolls (rotations) in realizing each of the wallpaper groups. But as you can see in the accompanying diagram, each frieze design in the wall hanging is built out of the same asymmetrical block of lace. The symmetries are formed by applying the transformations we want to that single motif. This is a very powerful technique, and it works with any motif that has no internal symmetries.

## Mouse symmetries

In the early stages of Mathemalchemy, when Dominique and Ingrid put out their call for stories to weave into the installation, I immediately thought that it would be fun to create a sort of symmetry scavenger hunt by scattering different symmetry types produced by a common motif through the exhibit.

I started sketching different motifs on my computer, but I was not having any luck finding a geometric block that was easy to spot, worked with different rotation angles, and gave aesthetically appealing designs. And then it occurred to me: what if instead of using abstract shapes, I made a recognizable form? Our collective imagination was already swirling with various woodland creatures. What about mice?

As the Mathemalchemy team mapped out our imaginary realm, the mice converged onto the central food source, the bakery. I was naturally interested in knitting the designs that could be knitted, and so I designed mouse wallpaper (!) for the side wall of the bakery. As you can see in the chart here, there are nine patterns on the wall, corresponding to the same nine groups as in *Crystalline*. The wall is a very slow knit on extremely tiny needles; at a rough estimate, the knitting time is about an hour per mouse.

There are three more wallpaper groups that don’t quite fit onto the wall because the stitches aren’t square. However, counted cross stitch embroidery uses a square grid, and our team has several practitioners. As the plan for the bakery’s neighborhood developed, we decided that the curio shop might need some decorative outdoor mats. Mary William and I adapted the mouse design to counted cross stitch, and Mary, Ingrid Daubechies, and Kathy Peterson set to stitching them. If you inspect each design, you should be able to spot some of the 90° rotational symmetries that exploit the squareness of the stitches.

Between the wall and the mats, we have accounted for twelve of the seventeen wallpaper groups. The five remaining groups fit into a hexagonal grid, a geometry often incorporated into quilts. Based on Mary’s sketches of various triangular, diamond, and kite-shaped mouse blocks, I worked up the pattern shown here. We will have this image printed onto fabric, and then Mary will painstakingly sew it into a quilt. The distinguishing feature of these wallpaper designs is that each has 60° and/or 120° rotational symmetries.

There will be a few more mischievous mice wending through the installation, but for those, you will just have to stay tuned!