August 19, 2013

Cross-Stitch Symmetry

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 D4 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. 


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.


Fay and Charlette said...

I love cross stitch and enjoyed this post very much! I am not a mathematician or anything like, that, just a regular person that sees math everywhere! My friends and kids think I am nuts! I told my pastor that I thought god was speaking to us thru math and he just stared at me.

migramorphs said...

You can bend cross-stitch rules by going off grid, or math rules by approximating.
I did the latter, only needing to approximate one point because all the rest are symmetrical.
(6,0), (3,5), (-3,5), (-6,0) etc. is close, and (8,0), (4,7) etc. is even closer.
I built a hexagon / triangle grid using (3,5).
It looks good to me, and the error isn't cumulative.
To me, the 5 patterns based on 60 and 120 look very possible in cross stitch.

migramorphs said...

The (6,0), (3,5) etc. pattern makes the minimum repeat pattern of 6 stitches (along the side of a "equilateral" triangle), as opposed to 2 for a square cross stitch. This is not much for a cross stitch pattern. I would be tempted to use the more exact 8-stitch repeat, with a little more room for the shape that repeats.

migramorphs said...

I did the first step here, but didn't actually try it. I see the problem now - the repeating shape will change its shape under some of the transformations.

One solution might be to only do a cross stitch at the vertices of the triangles, but I haven't tried this either. A very small mesh might help, and the stitch could be made big compared to the part that crosses.

Another possibility - I've seen hexagonal mesh fabric that could be laid over other fabric and used as a guide. This could also be used as a triangular guide because the hex centers form triangles.