3/24/2019 Hexagonal Cross Stitch – Isohedral

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Hexagonal Cross StitchMarch 23, 2019 | Filed under: Art, Bridges, Geometry, Research and tagged with: artdesign, artdesign_page, Bridges, crossstitch, mathematics | Edit

At least year’s Bridges Conference in Stockholm, I attended a short presentation by Susan Goldstineabout “self-diagramming lace”. As motivation for the new work she was presenting, Susanreferenced her paper from the year before on what she calls “symmetry samplers”. Samplers are anold tradition in fibre arts. A symmetry sampler combines small swatches depicting every possiblemathematical pattern type that can be realized in a given medium.

As an example, Susan showed a picture of Mary Shephard’s piece “Wallpapers in Cross-Stitch”:

Mary Shephard’s “Wallpapers in Cross-Stitch”

If you are familiar with repeating two-dimensional patterns, you will know that there are precisely 17distinct pattern types, known as wallpaper groups. This sampler shows only 12. But as Susanpointed out in her talk, Shephard’s piece is nevertheless a complete symmetry sampler. In cross-stitch, the stitches are applied to a woven fabric (called “aida”), which has holes for stitches

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arranged in a square lattice. Rectangular grids are fundamental to woven cloth, and necessarilyconstrain the symmetries that are achievable in cross-stitch. In particular, the square grid cannotsupport threefold or sixfold rotations (consider, for example, that no three points in a square latticecan form an equilateral triangle), which prohibits the five wallpaper groups that include them (for therecord, they are p3, p31m, p3m1, p6, and p6m in the traditional crystallographic notation).

But hang on a minute. The restriction to 12 groups follows directly from the structure of the aida, notfrom a mathematical law. And while it’s certainly true that almost every cloth on earth is woven usingwarp and weft at right angles to each other, producing a rectangular arrangement of holes, otherweaves are possible. I started thinking about an article I had worked on in my past capacity as editorof Journal of Mathematics and the Arts: Paul Gailiunas’s study of the mathematical properties ofmad weave:

Five mad weave polyhedral models by Paul Gailiunas, presented at Bridges 2013.

Mad weave is triaxial: it’s made from threads in three evenly spaced directions. And look at thearrangement of holes left behind:

A sample of mad weave with the holes illustrated on the right.

With a suitable stitching pattern, those holes would permit threefold and sixfold rotationalsymmetries!

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Sitting in Susan’s talk, these two threads came together (so to speak). If I could find (or make) asuitable piece of triaxial cloth, I could use it as a kind of aida. Then I’d have to develop a newstitching pattern that’s as analogous as possible to traditional cross-stitch. At that point, I’d be ableto create a hexagonal cross-stitch symmetry sampler. I resolved to attempt this project in time forthe following year’s Bridges conference.

Obviously, the cloth would have to be very fine, so that stitches on it would be sufficiently small. Iwasn’t eager to weave that myself by hand. So first, I looked into the manufacturing of woven triaxialfabrics. It turns out that they do exist. They have good engineering properties: they tend to be moreisotropic than regular woven cloth (they behave similarly in every direction) and less stretchy(because they don’t shear). After a lot of exploration, I reached out to Sakase Adtech in Japan. Theyvery generously sent me a few samples of their polyester triaxial fabrics. The weave they use isactually not the same as mad weave; it’s what Paul Gailiunas called “open hexagonal weave” in hisarticle. But the large hexagonal holes in that weave are still in the correct arrangement:

Image from Sakase Adtech’s site showing their triaxial weave.

I had never done any cross-stitching before, so I thought it would be prudent to start by creating atraditional square piece. I adapted a small piece of pixel art I had created a couple of years ago for alaptop sticker, leading to my first ever cross-stitch project:

3/24/2019 Hexagonal Cross Stitch – Isohedral

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My first ever cross-stitch project.

I made a bunch of mistakes in transcribing the pattern, and you should see the horrific mess in back.But it provided a suitable introduction to the medium, enough to get working on the hexagonalpiece.

I needed to develop a stitching pattern that would produce little units as similar as possible to the Xthat forms the units of traditional cross-stitch. After experimenting with and rejecting a fewalternatives, I settled on this pattern:

The three steps for stitching one unitin hexagonal cross-stitch: vertical, backslash, slash.

Here, three individual stitches in a prescribed order (“vertical, backslash, slash”) make up anasterisk-shaped unit. The stitches begin in the six hexagonal holes surrounding another hexagon andjump over that central hole, meaning that a number of holes have no stitches in them. That’s fine, of

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course, if the units are small enough, and this approach turns out to be the most convenient way towork with this fabric.

It was then time to design patterns. I decided to divide the sampler into a kind of hexagonalhoneycomb (to emphasize the sixfold symmetry), and I created vector drawings in Illustrator forevery cell. Each drawing shows the low-level hexagonal holes of the fabric, with larger, filled-inhexagons showing the intended units:

An example of a piece of hexagonal cross-stitch. Click to

download a vector illustration as a PDF.

With all those pieces in place, the last step was very simple: hours upon hours of punishing, tediousmanual labour. My close-up vision has taken a sharp nosedive in the past couple of years, so here Iwas aided by a very stylish magnifying visor with built-in LED headlamp. I’m quite happy with thefinished result:

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A (more) complete cross-stitch symmetry sampler (2019). Click to see a full-size image.

There are five large hexagonal cells, containing the five symmetry groups that are not possible usingtraditional cross-stitch: reading in rows from the top, p31m, p3m1, p3, p6, and p6m. The other threecells are each divided into three rhombi, showing the other nine wallpaper groups that are alsopossible with hexagonal holes: p1, p2, pm, pg, cm, pmg, pgg, pmm, and cmm. This fabric can’tshow fourfold rotations, which leaves precisely three wallpaper groups (pm, p4g, and p4m) on thesewing room floor. That makes for a grand total of 14 groups, versus 12 for traditional cross-stitch.Take that, Mary Shephard!

A pedantic mathematician will no doubt observe that the sampler doesn’t actually have all thesymmetries advertised here. In each individual unit, the stitches are layered, which strictly speaking

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eliminates nearly all possible kinds of symmetry. Pedant, I salute you! Indeed, if you take the layeringof threads into account, and require every unit to follow the vertical-backslash-slash pattern, then Ithink you’re able to represent only groups p1 and p2. Of course, the same layering problem ariseswith traditional cross stitch, and we respond with the same simplification: for the purpose ofevaluating symmetry we flatten stacks of thread into conceptual blobs of colour.

Very special thanks to Ryoji Sakai and Sakase Adtech for sending me samples of their triaxialfabrics, and to Teruhisa Sugimoto for providing assistance in reaching out to them. Thanks also toVeronika Irvine for providing feedback on both the mathematical and stitching sides of this project,and to Linda Carson for suggesting the oddly shaped matte, which was expertly cut by The Artstorein Waterloo. And of course, thanks to my wife Nathalie for teaching me the basics of cross-stitch,giving me access to her large stash of supplies, and not mocking my visor too much.

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One Response to "Hexagonal Cross Stitch"

Marlene Knell says:March 24, 2019 at 8:12 pm   (Edit)

Wonderful such inspiration thsnk you

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