Étude no. 15: permutations

I’m designing a pattern which needs a little simple lace in a multiple of 3 stitches at the beginning. I didn’t want a secret code lace. I wanted a multiple of 2, 4, 6, or 12 rows. In the end, I used simple diagonal lines because it flowed nicely with the other lace in the design. But I was struck by the scarcity of basic lace patterns in multiples of three in the places I was looking..

Suddenly it occurred to me that with only three stitches in a row, the most basic lace would have one knit stitch, one decrease, and one yarnover per row. This essentially makes for six permutations for a given row. (I simplified matters by pretending that all decreases are the same.)

six permutations of three stitches

Mathematically speaking, this means that there ought to be 6 possible two-row lace patterns with these three stitches (including a plain alternating row), 36 possible four-row patterns, and 216 possible six-row lace patterns. Obviously they won’t all be nice, though fiddling with changing which way the decreases lean helps a lot.

Also, a fair percentage of these are essentially  duplicates. Compare this pair:

basically duplicates 1

When repeated, the effect will be basically the same, as will this one (though it’s a different effect):

basically duplicates 2

All four of these charts are part of the 216 permutations.

I didn’t feel like knitting all of them, but I thought it would be fun to generate a few stitch patterns using the six possible rows and a six-sided die. Maybe you’d like to try doing it too? I recommend reading through my process, though – there’s a structural concern I didn’t think of when I started this étude.

General Notes:

  • These stitch patterns such as might be found in a stitch dictionary. They are not patterns for finished objects. You will need to add selvedges or some other form of knitted stitches to either side.
  • Designers, please feel free to use these stitches in your patterns. I’d like credit but won’t be offended if people don’t give it.
  • If you like my posts like this, please consider supporting me on Patreon or donating with my Paypal tip jar in the sidebar. Thanks!

Abbreviations:

  • k: knit
  • k2tog: knit 2 stitches together as if they were 1. (Right-leaning decrease)
  • ssk: slip each of the next 2 stitches as if to knit, then knit them together through the back loop. (Left-leaning decrease)
  • yo: yarnover

I decided to start with a six row pattern; this meant picking three lace rows at random. The dice gave me 4, 4, and 6.

Permutation 1 of 3-stitch

permutation 1

Notes:

Row 1 (RS): *k2tog, k1, yo, work from *.
Rows 2, 4, 6: purl.
Row 3: *k2tog, k1, yo, work from *.
Row 5: *yo, k2tog, k1, work from *.

I like this stitch pattern, but I realized pretty quickly that my knitting was biased: it was leaning slightly. This is helped by blocking, but I wouldn’t want to count on the bias not returning. To avoid this, I needed to have an even number of lace rows, half of which had the decrease to the right of the yarnover, and half of which had the decrease to the left.

I decided to knit this stitch pattern from the text, omitting the wrong side rows. This makes this a six row pattern with the decreases and increases spaced in such a way to eliminate the bias. I like this a lot.

Permutation 2 of 3-stitch lace

permutation 2

Notes:

  • There is lace every row, but I don’t think it’s a particularly difficult example of the category.
  • It’s reversible.
  • I’ve made a stitch map for this stitch.

Row 1 (RS): *k2tog, k1, yo; work from *.
Row 2: *k2tog, k1, yo; work from *.
Row 3: *yo, k2tog, k1; work from *.
Row 4: *k2tog, k1, yo; work from *.
Row 5: *k2tog, k1, yo; work from *.
Row 6: *yo, k2tog, k1; work from *.

I realized that another way to make the first version stop biasing was to mirror it, even though it made it a six stitch repeat and added double yarnovers.

Permutation of 3-stitch lace

permutation 3

Notes:

Row 1 (RS): *k2tog, k1, yo x 2, k1, ssk; work from *.
Row 2 (WS): *p2, k1, p3; work from *.
Row 3: *k2tog, k1, yo x 2, k1, ssk; work from *.
Row 4: *p2, k1, p3; work from *.
Row 5: *yo, k2tog, k2, ssk, yo; work from *.
Row 6: *p5, k1; work from *.

I think this one has some promise, but I think I’d want to play with more than just the decreases before I’d be content.

I decided to try another set of numbers next, being careful with the bias issue. I rolled these four numbers: 4, 2, 6, 3. Two have YOs to the right, and two to the left, so that’s all right.

Permutation 4 of 3-stitch lace

permutation 4

Notes:

Row 1 (RS): *k2tog, k1, yo; work from *.
Rows 2, 4, 6, 8: purl.
Row 3: *k2tog, yo, k1; work from *.
Row 5: *yo, k2tog, k1; work from *.
Row 7: *yo, k1, k2tog; work from *.

After knitting the swatch, two things came to mind: first, mirroring the rows on the second time through, and second, editing the decreases. I decided to start with mirroring the rows.

The first thing I noticed is that this made some of the rows into duplicates (highlighted in the second chart). There’s nothing inherently wrong with this—some stitch patterns do this—but I decided to get rid of one each of the duplicates before knitting the next swatch.

Permutation 5 of 3-stitch lace

permutation 5

Notes:

Row 1 (RS): *k2tog, k1, yo; work from *.
Row 2 and all even rows: purl.
Row 3: *k2tog, yo, k1; work from *.
Row 5: *yo, k2tog, k1; work from *.
Row 7: *yo, k1, k2tog; work from *.
Row 9: *k1, yo, k2tog; work from *.
Row 11: *k1, k2tog, yo; work from *.

This is getting closer to something I like, but it’s not quite there. I changed three of the decreases to ssk, and that made me much happier:

Permutation 6 of 3-stitch lace

permutation 6

Notes:

Row 1 (RS): *k2tog, k1, yo; work from *.
Row 2 and all even rows: purl.
Row 3: *ssk, yo, k1; work from *.
Row 5: *yo, ssk, k1; work from *.
Row 7: *yo, k1, ssk; work from *.
Row 9: *k1, yo, k2tog; work from *.
Row 11: *k1, k2tog, yo; work from *.

I think this is pretty good. There’s a few tweaks I might want to try later: stitch 3 from row 3 and stitch 1 from row 9 are kind of niggling at me. What would happen if I beaded them? Or twisted them by knitting them through the back loop?

This might all give you some ideas for how to play with permutations yourself; I hope you do!

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3 thoughts on “Étude no. 15: permutations

  1. I like this idea. Usually when I’ve made up lace it’s been a picture. So my version of pine trees our cats paw.

    I wonder if the difference is due to the way we see or not see images in our heads.

    1. Possibly? I do sometimes pull out motifs that look like images from my lace. But I think the underlying thing is that I enjoy using mathematics or letters as a way of generating the charts that then need editing.

      I do draw things, and I do plan on designing pictoral lace on a hexagonal mesh at some point.

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