The craft of counted
cross-stitch lends itself to the creation of elegant patterns on fabric,
and mathematician Mary D. Shepherd
has taken advantage of this form of needlework to vividly illustrate a wide
variety of symmetry patterns.

*Mary Shepherd with her cross-stitch symmetries sampler.*

Photo by I. Peterson

The fabric that Shepherd uses is a grid of squares, and the
basic stitch appears as an X on the fabric. In other words, one cross-stitch covers
one square of the fabric.

Stitching over squares constrains the number of symmetry
patterns that you can illustrate using this technique. The reason for this
constraint is that the only possible subdivision of a square is with a stitch
that "covers" half a square on the diagonal. In effect, a half
cross-stitch splits a square into two isosceles triangles, covering only one of
the triangles.

This means that the only angles you can create in a counted
cross-stitch pattern are multiples of 45 degrees.

Wallpaper patterns have translations in each direction along
two intersecting lines. Of the 17 possible wallpaper patterns, only 12
can be done with a combination of cross-stitches and half cross-stitches. The
other five patterns involve angles of 60 and 120 degrees, and so are not
possible in counted cross-stitch.

*Six of the 12 wallpaper patterns that can be done in counted cross-stitch needlework. Top row, left to right: p1 (translation only), pg (glide reflection), pm (glide reflection axis along line of reflection). Bottom row, left to right: cm (glide reflection axis not along line of reflection), p2 (180-degree rotation), pmm (reflection and 180-degree rotation).*

Courtesy of Mary D. Shepherd

Shepherd has also worked on both frieze and
rosette symmetry patterns. Frieze patterns, often used for borders, have
translations in two directions. A rosette pattern has at least one point that
is not moved by any of the symmetry transformations (translation, rotation, reflection,
and glide reflection), Shepherd notes. Hence, the only transformations that can
occur in rosette patterns are reflections and rotations.

Rosette patterns, for example, give a nice visualization of
the symmetries of a square (technically,
the group D

_{4}and all its subgroups), she says.*Rosette patterns for visualizing the symmetries of a square (the dihedral group of the square).*

Courtesy of Mary D. Shepherd

Shepherd provides instructions for crafting a "symmetries sampler" in the book

*Making Mathematics with Needlework: Ten Papers and Ten Projects*(A K Peters).
She has also used counted cross-stitch examples in the classroom to illustrate and explore ideas about symmetry groups and subgroups.

**Reference**:

Shepherd, Mary D. 2007. “Symmetry Patterns in Cross-Stitch.”
In

*Making Mathematics with Needlework: Ten Papers and Ten Projects*, sarah-marie belcastro and Carolyn Yackel, editors. A K Peters.