Social Mathematics

written by Shannon Paper-Negaard

Knitting and Math

I have two main communities: my math department (I’m a grad student) and my knitting group. When I talk to some people in the knitting community at large, they have the impression that more complex knitting is very “mathy,” because “you have to count stitches” in various patterns.

I don’t really see the counting as mathematical. The person who designed the original pattern used spatial reasoning, and that’s math. When I knit complicated cable or lace patterns, I visualize the pattern part of it, and that’s math. Even something like adjusting gauges (the number of stitches per inch) to make garments of an intended size – that is sometimes math.

And a particular type of yarn that I’m enamored with – so much so, that I’m writing this – is mathematical.

Getting Stripes

For the non-knitting folks: if you’ve ever seen a hand knitted garment with multiple colors, there are a few ways for a knitter to achieve this. The first (more traditional) way is to have several balls of yarn and to knit each stitch with one of these, changing as one goes. That’s not a bad way to do it, if one only has to change colors a few times. But there’s a catch. Every time one length of yarn is broken to let another be tied in, there are now two ends to weave in. If you don’t knit or crochet, ask someone who does how they feel about weaving in ends. You’ll soon learn it’s not the enjoyable part of fibre art.

Another project, with many ends, waiting to be woven in.

The second way to change colors is to buy a ball (or skein or hank) of yarn, which already has multiple colors in it.

Usually, when a ball of yarn has multiple colors, those colors occur at regular intervals. If the yarn was dyed as a 2 yard skein, then every two yards, the same color will appear. So, roughly the same number of stitches will be required to get back to a color.

This makes for a really pretty fabric, if you’re knitting something that has about the same width (or circumference) throughout the project. Socks, mittens, rectangular scarves, and even hats will all look roughly as you would expect them to.

These socks – these socks look great in a yarn that works this way.

However, a lot of the patterns knitters like to make change width as they go. Specifically, triangle shawls and semicircular shawls have been extremely popular in recent years. Here’s how they’re constructed.

We cast on just a couple of stitches (usually 4 or 6) to start and then knit back and forth (as indicated by the very “high-tech,” penned arrows above). Each row, we add a few stitches. By the time we get to the outside edge, there are hundreds of stitches in each row.

Now, let’s say our skein had fairly long bands, so we actually get full color stripes all the way to the end of the shawl. In knitting, length of yarn translates to a number of stitches, and stitches translate to area: a block of 100 stitches can be 2 rows of 50 stitches, or it can be 10 rows of 10 stitches, or it can be a single row with 100 stitches. It all depends on the row width. So, if 3 yards of yarn gives 100 stitches, the area of knitted fabric won’t change – but the shape it takes might. Knitting friends, this spatial reasoning was brought to you by mathematical thinking J. You’ve all done math, every time you’ve estimated the yarn needed for a pair of socks or a scarf. No counting needed.

Anyway, in the shawls pictured above, we can clearly see the row sizes are changing and it doesn’t take much mathematical intuition to realize that a stripe that’s 1” deep near the start (the part with short rows) has a lot less area, and therefore a lot fewer stitches, than a 1” stripe near the end (with the long rows). So it takes much less yarn to make that 1” stripe near the center or beginning, than it does to make a stripe of the same width toward the outer edge.

How much difference, exactly? Caterpillar Green Yarns figured that out, and we just get to reverse engineer it.

Finally, the mathy yarn

The shawl below – before Caterpillar Green’s innovation – could only have been constructed by

• purchasing skeins of all of the unique colors you see below,
• knitting a few rows in grey, cutting the yarn, tying on purple yarn,
• knitting a few rows in purple, cutting the yarn, tying on grey yarn,
• knitting a few rows in grey, cutting the yarn, tying on red yarn,
• etc,

and then weaving in all of those ends. Again, weaving in that many ends just sucks.

But Caterpillar Green found a way to dye the yarn – all in one skein – with the appropriate lengths of yarn.

Our goal is to understand what lengths and why it works.

We’ll use the first style of triangular shawl as our model. Now, we need a standard unit for the length of our yarn. Let’s use the length it takes to make that first “stripe.” If the whole picture shows the shawl, the darkened triangles, together, make up the first stripe. It’s actually going to be more convenient to think about half of that stripe.

(Yes, the first stripe looks like two little triangles put together. Further on, they’ll actually look more stripes.)

Also, every knitter will make a slightly different size triangle with the same length of yarn, and those will change if they change needle sizes. But most knitters also stay consistent within a project. The same knitter will also be knitting the later rows, so the amount of yarn needed for an equal area will remain the same.

Let’s break the rest of the shawl up. This is somewhat simplified. I only have 4 color stripes, but we’ll get the idea. The sides are also mirrored, so I can show off the stripes and the triangles that are key.

The second stripe is made up of 3 of those triangles, or 1+2. The third stripe has 5 triangles, or 3+2. In fact, if we keep adding two triangles for each new section, we have enough area to all stripes in the pattern. Also, we’re using all the odd numbers.

Doubling that (to account for both sides) doesn’t change the proportions. The second full stripe (both sides together) contains 3 times the area as the first (the two dark triangles put together). So the second color section in our yarn has to have 3 times as much yarn as the first. The third color section has to have 5 times as much as the first. The forth has to have 7 times as much as the first, etc.

This is just one shape, though. Let’s see what else can be knit with this yarn.

The key is to notice that the total amount of yarn – using that first color section as our unit – is as follows:

After 1 color: 1 = 1²,

after 2 colors: 1 + 3 = 4 = 2²,

after 3 colors: 4 + 5 = 9 = 3²,

after 4 colors: 9 + 7 = 16 = 4²,

…

after n colors: n².

It’s all squares. Or, at least, it’s perfect squares times the yarn that was in that original color section.

It turns out, there are many shapes, for which this same pattern will make nice color stripes. A few of them are drawn below.

Another triangular shape

A circle of radius r, which would have r color sections, is r². The first ring (actually a small circle) uses the standard amount of yarn in the first color section of the yarn. It happens to make a circle of radius r, with arear². The next section has three times as much yarn, so it adds an area of 3r². The total area is now 4r² =(2r)², or a circle of radius 2r, which is exactly what we wanted. It means we have a ring around the first circle, of width r. It’s easy to see that adding the next color strand, which contains 5 times as much as the first, adds 5r² in area, for a total of 9r² =(3r)², or a circle of radius 3r. The pattern continues. Even though we can’t break the subsequent rings up, visually, into smaller circles (like we did with the triangles), the areas still work as we would hope.

Similar calculations work for a semicircular shawl or even a quarter circle shawl. They also work for squares, different shaped triangles, and wedges of triangles. The yarn, which worked so well for a triangular shawl, also lends itself well to a variety of shapes. Some example shawls, from the Caterpillar Green Yarns website, are shown below.

Here is a photo of the shawl I designed. It looks a little less triangular, because of the way the increases curve.

There are so many things that can be done with this yarn. I wrote a pattern for it.  Of course, the makers of the yarn at Caterpillar Green Yarns will happily give suggestions. Then, there’s the community at Ravelry, who have found wonderful things to do with a yarn that stripes in a different way than we’ve all been used to.

There aren’t crochet projects yet, but who knows, maybe you’re the industrious crocheter to come up with an equally lovely project in your craft.

If you are a knitter, be sure to check out Ravelry, even if you’re not searching for this yarn. It’s where I get inspiration for almost all of my knitting projects, and when I’ve finished with them, it has been a wonderful place to display pictures. I only post about 70% of what I complete, but that percentage is getting better, especially as it gets easier to upload photos.

I hope you liked learning about this yarn. I’ve already made two lovely shawls and have a third set aside as a fall project. And of course, I shall only work on it after putting in a full day of work on my research and grading (wink).

Author’s bio: Shannon Paper-Negaard has a masters and is working toward a PhD, both in mathematics. She is studying dynamical systems at the University of Minnesota. She loves to knit and spin yarn, and she’ll talk your ear off about knitting, math, travel, chocolate, or her bicycle. Find her on Twitter at @ShannonPaper or on Ravelry as paperdolls.