Optimization

When I buy yarn that isn’t already in my stash, I feel a lot of compulsion to come up with a plan for it and execute that plan promptly. As it is, I have enough yarn to last me 20 years after I die, so if I’m going to purchase yet more yarn, it needs to be processed pretty quickly so that I don’t have a backlog of unused yarn sufficient for 30 years after I die. But I do buy yarn when my LYS (local yarn store, knitter jargon), Lovelyarns, has a trunk show, because I want to support Lovelyarns and local makers: the lesson being, if you like having an LYS, buy your yarn there. Don’t walk in, look at the yarn, and buy it cheaper online, not if you want to continue to have an LYS. So at a recent trunk show by Bad Wolf Girl Studios, I got two sweater quantities (squees, in knitter jargon). One was four skeins in colors to make a yellow-to-pink gradient. The other was four yarns, three in a dark blue-green and one aqua. I texted photographs of the yarn to my two grown daughters, each in distant places, and very quickly it was decided that the yellow-to-pink gradient would go to my older daughter and the dark blue-green and aqua would go to the younger.

Two squees of Bad Wolf Girl yarn destined for my knitting machine and my two daughters

I had a vision for each squee before I even pulled out my credit card. I had just completed my first two machine-knit raglans and wanted to reinforce what I had learned about the math and execution of that style, so both would be raglans. The yellow-pink gradient would be a fade of each yarn blending into the next colorway in the series, and the yoke would be 10-row stripes of different but related colors left over from the machine-knit raglan I had made for myself, with accents in a limey, leafy green. The only request my daughter, a willowy, elegant young woman, made of me was to be sure the sweater was long enough to cover her fly for those times when she forgets to zip it. She has an unusually long torso, so that meant that the length of that sweater would be like a dress for me. This was going to take a lot of yarn, but I wasn’t worried about running out. She’s long but narrow; the four skeins came to 400 grams of yarn, which is plenty for a person of my girth and slimmer, which both of my daughters are; and the yoke would be made of additional yarn.

A comfortable circumference for the sweater would be 40″, and 28″ from nape to hem ought to cover even my daughter’s endless torso, so I made a swatch and calculated the number of stitches needed for the 20″ width (140 stitches) and rows for the 28″ length (198 rows of fade sequences, plus 10 for the folded hem). The yoke was going to be a rota of five colors in 10-row stripes with a depth of about 8″, so I needed a number of rows that ended in a zero in order to end in a full repeat of a 10-row stripe. I settled on a depth of 90 rows, which was a bit deeper than 8 inches. The start of the yoke was at row 208, and I devised the fade rota to extend from the top of the hem up to that row. The rota was: 26 rows color A; 3 rows A, 1 row B, twice; 2 rows A, 2 rows B, twice; 8 rows alternating A then B; 2 rows A, 2 rows B, twice; 1 row A, 3 rows B, twice, and so forth for each of the four different colorways. The color transitions were based on 8-row segments. Changing the colors so often was fiddly and required a lot of attention to keep track of where I was and was not to be done when I was tired or distracted, but I managed to stay more or less accurate by keeping a close eye on the stitch counter and keeping a running tab on the calculator of my phone, adding 8 to the previous number every time I finished a segment, unless I reached one of the long single color segments (what a relief!), when I added 26. I didn’t put any shaping into the body, just knitted it straight. My daughter’s body provided the shaping. It also wasn’t necessary to put short rows into the upper back, as I did for my husband’s raglan and my own. Straight young backs don’t require that accommodation.

The numbers for this sweater required more precise calculations than Elizabeth Zimmermann’s EPS/Elizabeth’s Percentage System, which I discussed in my Trust the Numbers! post. EPS is a very simple formula that works well for typical body proportions when yoke patterning can end anywhere or when the yoke patterning happens to work for the formula. However, my daughter’s arms are slimmer than the sleeve circumference EPS would have given me, and the striping sequence came out to a yoke depth of 90 rows. Then I calculated the number of stitches I needed to reduce. I started out with 140 stitches on the front and back, and put 10 stitches onto waste yarn on either side of the piece for the base of the armhole, leaving 120 stitches. Meanwhile, the sleeve at its widest was 92 stitches, reduced to 72 stitches for the corresponding stitches at the base of the armhole. I wanted to make sure I had 4 stitches at the end of the sleeve’s raglan decreases, two for the seam and two for a tiny bit more space in the neck, so that meant decreasing 68 stitches. Decreasing 68 stitches in the sleeves meant decreasing 68 stitches each for the front and back, leaving 52 stitches for the neckline for those pieces, which would create a comfortable neckline. So I needed to decrease 68 stitches over the course of 90 rows. I put my numbers through the Magic Formula, also explained in the Trust the Numbers! post, and arrived at numbers for a raglan slope that gave me two rates of decrease, almost evenly divided over the length of the yoke, decreasing 2 stitches every other row 23 times (2×23=46 rows) and 2 stitches every 4 rows 11 times (4×11=44 rows).

My version of a pattern

Then I did an experiment that turned out all right, but I’ll never do it again. The conventional wisdom is to work the slower rate of decrease first, at the bottom of the raglan, that is, the wider part of the slope, then the faster rate of decrease. But I had somehow gotten the idea that that would produce an outward directed raglan line that would be less attractive than an inward directed line. So I did the faster rate of decrease first, then the slower rate of decrease on the back, and then geometric reality set in: I was making the wider part of the back narrower and the narrow part of the back wider.

My challenge to the orthodoxy of raglan shaping. I fought the orthodoxy and the orthodoxy won.

I didn’t want to frog (knitter jargon for rip it, rip it) because I had broken my yarn for nine 10-row stripes and it’s a pain to reconnect short lengths of yarn for reuse on the knitting machine, so I decided to knit the front the same way as the back and reverse the rates of decrease on the sleeves to average out the rates of decrease. Then I joined the raglan with a visible, decorative seam in contrasting green, to disguise the differing appearance of the two sides of the seams, and also because the green visible seam was a design element that connected the look of the green hem, cuffs, and neckband with the diagonal lines of the raglan.

Reversing the rate of decrease on the sleeve mitigated the error of attempting the fast rate of decrease first on the front and back, but I won’t do that again.

The sweater fits my daughter fine, despite my error in judgement. But I have learned that there might be good reasons for conventional wisdom, at least in this case.

All’s well that ends well, thanks to my daughter’s slender body and the forgiving nature of knitting, but that raglan slope is a lessons-learned.
Back view. That raglan seam looks pretty good, despite my regrets.
Green seams on display

My idea for the sweater from the other squee was to have a striping sequence based on 18 rows of dark blue-green (a colorway descriptively named Potions Master) and 2 rows of a tropical aqua whose colorway name was Siren, with the colors reversing in the yoke in a transition whose length and placement needed to be determined by means of more math than I know. Since I had three skeins in the one color and one skein in the other, I needed to make sure I was using the two colors in a 3:1 ratio, and math is the superpower that enables you to know how to keep from running out of one color or another.

One of the regulars at Lovelyarns, a young man named Evan, is a Ph.D candidate in the physics department at Johns Hopkins University, and he knows his math as well as his knitting. I told him my plan and asked him if I was going to run out of either color. He took vital statistics, such as the weight and yardage of my yarn and the dimensions I was seeking from the finished garment, and told me he’d get back to me. While I waited to hear back from him, my vision of the design began to clarify. I wanted a raglan pullover in which the blue-green dominated over the aqua at an 18:2 ratio in the body and arms, then transitioned at the yoke to a 2:18 ratio, aqua dominating. Intuition and experience gave me the sense that, for a sweater of ordinary proportions, armpit level is approximately the border between 3/4 of the garment below that level and 1/4 above it, and I was prepared to go ahead with this theory if I didn’t hear back from Evan. But he came back a few days later with five pages of sketches and mathematical formulae. I was impressed with the long stream of letters, numbers, symbols, and parentheses, perversely pleased that my seemingly simple question was so complicated.

The other two pages of Evan’s notes and calculations are on the backs of two of the pages

We talked some more about the various transitional segments and how they would look on a body. I explained that I wanted the transitional segments to be based on a 20-row repeat, because that’s easiest for me to keep track of when I’m operating the machine. We decided that the best place to start the transition from 18:2 dark-green:aqua to 2:18 aqua:dark-green was slightly below the base of the armhole. We weighed and measured my swatch. More sketches, more math, and in the end we came up with numbers to which he could give his blessing: I should have enough of both colors on the basis of the plan we arrived at.

This sweater, like the first, would be based on a cast-on of 140 stitches knitted straight up to the armholes. My 20-stitch rota had me knitting 9 full 18:2 repeats (180 rows) after the 10 rows of the folded hem, and then an additional 10 rows before the start of the yoke, with 10 stitches per side put on waste yarn for the base of the armhole. These 10 rows were the start of the transition to 2:18, which consisted of a 14:6 stripe pair, 10:10 pair, and 6:14 pair before getting to the final two stripe pairs at 2:18 with aqua dominating. That is, after the final 10 rows of the body in blue-green, I knitted 4 more rows of that color at the start of the raglan decreases, then progressed through the transition to the final two sequences of mostly aqua. Once again, the depth of this yoke was 90 rows, like the yoke of the previous sweater. That meant that I could use exactly the same numbers as my older daughter’s sweater for the raglan shaping, but this time I did the slower rate of decrease first, then the faster rate of decrease. It was perfect, the poster child for knitting the two rates of decrease the conventional way. The decreases aligned perfectly at the seams, so there was no need to cover the seams with a 3-row visible seam in order to disguise unmatching decreases, but I did it anyway because it was a design element that emphasized the angles of the raglan and reinforced the impact of the froggy green contrast color.

Slower rate of decrease, then faster, is the way to go in raglan shaping. I have learned my lesson.

So I completed the sweater without a whole lot of yarn to spare, but there were no yarn emergencies or even scary games of yarn chicken. I should note that I took measures to avoid this by using Russian joins to connect the start of one cake of yarn to the cake that was running out and continuing to knit with the joins in the middle of the row, which conserved quite a lot of yarn. When I weighed the two colors of leftover yarn, I discovered that I had 36 g remaining of the blue-green and 11 g of the aqua, close enough to 3:1, which showed that my design had succeeded in using the two colors at the 3:1 ratio I needed.

When the sweater was finished, I sat down with Evan and asked him to explain the math to me again. He declined to teach me advanced math and engineering as we sat at the back table of Lovelyarns, but he said that the five pages of calculations that brought him to green-lighting the configuration we arrived at is why mathematicians get paid the big bucks. He explained that the real-life, practical situation I had presented him with is what the quantitative sciences, particularly engineering, call an “optimization problem”. Optimization is the way you take a bunch of different parts of a process that are all doing their own thing and combine them to work together for a unified purpose in a way that doesn’t cause the whole thing to blow up. In the case of this optimization problem, the different parts were the yarn usage in the simple rectangle of the 18:2 body segment, the trapezoid of the sleeves at the 18:2 ratio, and the trapezoids of the yoke at the 14:6 ratio, 10:10 ratio, 6:14 ratio, and 2:18 ratio. The math required to figure out the ratio of the yarn usage was algebra and geometry, but I forgot my algebra decades ago and never actually learned geometry when I took it in high school. My intuition was that I would get a 3:1 ratio by transitioning from mostly blue-green to mostly aqua between the armpit and the widest part of the shoulder, but if I needed to be sure ahead of time that my intuition was accurate, my level of math skill would have required me to unravel a row of my 50 stitch/50 row swatch, measure the length of the unraveled yarn, and figure the width and height of a stitch, then do a lot of adding and multiplying to get the numbers for how much of each color I would use. Math, however, cleans up the messiness of concrete numbers and pares it down to abstract ratios.

But for now, I will have to either go with my intuition about the amount of yarn I need and resign myself to playing yarn chicken, or unravel swatches and add up the number of stitches needed to make trapezoids until I derive the concepts of geometry that eluded me in 10th grade. Evan did promise to give me a cleaned-up version of those five pages of sketches and formulae with a linear explanation of his thought process. So maybe he actually will succeed at teaching me higher math and engineering while sitting at the back table of Lovelyarns!

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9 thoughts on “Optimization

    1. I don’t think I have anything helpful to say about how I chose those colors. I didn’t think about it, I just knew the moment I saw the yarn. Although orange and lime green work with an amazing range of colors and make you look like a color genius.

      Like

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