Miriam Felton

I’ve begun the Aspen Grove shawl for Wooly Wonka Fibers. Last year most of my shawls were top-down triangles, and this year there have been a lot of stoles (although you haven’t really seen any of them, trust me). So for this one I wanted to do something different. Maybe something half-round.

First I e-mailed my math genius of a sister about formulas to calculate how often I should increase for a half-round shawl. She sent me this in response. If you’re not mathy inclined, maybe just skip it, but for those who are curious, here it is. It’s long:

The “wiki(circumference)” (half-circumference actually) of a half-round shawl is C = pi * r where r is the radius in inches. The increases you have to do are the rate of change of this circumference with respect to increasing radius, dC/dr if you remember your calculus. But r, the radius, increases as you work with respect to the number of rows t and the height of the stitches b (b being calculable from your row gauge swatch; see below); r = bt . So using the chain rule, dC/dr = pi dr/dt = pi * b. Please note that when the row gauge is constant, this is a constant rate of change; you will not have to add an increasing number of stitches as the rows get larger. If the row gauge is not constant, if you were to increase stitch size, say by gradually changing needle sizes over the course of the shawl to go from tiny needles to very large ones, you’d have to make a larger number of increases over the increase rows as the needle sizes got larger. This is odd, but true.

Now let a be the width of the stitches in inches (calculable from the gauge swatch). Pi * b gives you the number of inches you need to increase each time, so pi * b/a gives you the number of stitches that need to be increased each row. Notice that b/a is the aspect ratio of the stitch. The reason that increasing about 6 stitches each row produces a circle is that the aspect ratio of the knit stitch, like that of single crochet, is a little less than 1. If you had taller stitches, e.g. double crochet stitches, you’d have to increase quite a bit more per row to get it to lie flat.

To sum up, in case you weren’t able to follow all of the above:
a = width (in inches) of stitch
b = height (in inches) of stitch
number of stitches to increase each row = pi * b/a

Sample calculation: suppose you work up a gauge swatch and find you get 24 stitches and 32 rows to four inches. Then a = 4/24 = 1/6 and b = 4/32 = 1/8, so the number of stitches to increase each row is pi * (1/8) / (1/6) = 3 pi/4 = 2.705 approximately. So if that is your gauge, you’d have to increase *on average* 2.705 stitches per row.

Now the hard part is working out where to put these increases so that they average 2.705 stitches per row. We can only increase whole numbers of stitches per whole number of rows. If we increase, say, 3 or more stitches per row on average, we will end up with a shawl with a hyperbolic (fluttery) surface, and if we increase less, we will end up with an elliptical (convex) surface, neither of which will lie flat. If it’s only slightly hyperbolic or elliptical, we can make it more or less right when we block the shawl; but if it’s really off, no amount of blocking will help it. The larger the shawl, too, the more precise we need to be. A slight deviation from flat will produce a much more dramatic effect over a large piece than over a small one. (Note: if you want a fluttery shawl, do more increases than what your calculation suggests.)

Also, in real knitting patterns, even if we could increase exactly 2.705 stitches per row, we wouldn’t want to. We like to have increase rows and pattern rows, and moreover we want to have increase rows always worked on the same side of the knitting so that we can get a nice pattern going. Also we will want to keep the increase rows from being too far apart– working all the necessary increases in the last row isn’t going to produce a smoothly increasing shawl! The farther apart the increase rows are, the more of a “tiers of ruffles” effect you’ll get. If that’s what you want, go for it! but if not, make the increase rows closer together. (Also it goes without saying that you will want to spread the increases as evenly as possible across the increase row.)

Mathematically, this is equivalent to finding a rational number (i.e. fraction) with a relatively small, even denominator that approximates the number of increases per row, and this rational number has to be close enough to the actual value that any errors in it can be taken up in blocking the shawl. The actual way you end up doing it will depend a lot on the pattern stitch you are using. The easiest way to do it is to decide on the pattern stitch you want and then calculate the number of increases per pattern repeat. So for example if you use a pattern stitch that repeats every 10 rows, you might want to use a denominator of 10 and solve for the numerator, like so:

pi * b/a = 2.705 = x/10
x = 2.705 * 10 = 27.05 or approximately 27

so if you use a 10 row pattern stitch with the gauge specified above, you will need to increase 27 stitches in that 10 rows. You can do it all at the beginning, all at the end, all in the middle, half at the beginning and half at the end, or work it in sneakily during the pattern, it is up to you.

If you want to do bands of different pattern stitches of different numbers of rows, you can calculate it separately for each band. So for example, let’s say that after you did the 10-row pattern stitch, you switch to a 16-row pattern stitch. Then you have:
2.705 = x/16
x = 16 * 2.705 = 43.28 or approximately 43 stitches
so during that band you will have to increase 43 stitches.

Now you may have noticed that in both of these band calculations the number of stitches to increase was rounded down. If you find that you are constantly rounding down, your number of increased stitches will be too small, so keep track of these rounding errors. If you round down 43.28 to 43 over the course of four bands, you’ll be short a stitch at the end. One stitch over the course of an entire shawl is nothing, but if you work a large series of 16 row bands (say you have a tiny gauge) you could be short 10 or 12 stitches at the end, this is not a good thing. I don’t know how large or repetitious you intend the shawl to be and what the gauge would be, so in all probability the rounding errors will add up to close to nothing, or cancel each other out ( i.e. if you had to round one band’s increases up it compensates for rounding the other band’s down). But it’s something to keep an eye on as you design. You may have to throw in a “leap stitch” every now and then. It works the same way as leap years– we throw in an extra day every 4 years, take out a leap day every 400 years, etc.

Another design consideration may be that you would want the shawl to be flattish in the middle and fluttery (hyperbolic) at the edge. In that case you would work the body of the shawl in band(s) with the correct number of increase stitches, and work the final band(s) with a too-large number of increase stitches.

In the event that you have a pattern stitch that you can repeat over any number of rows and it would be convenient to work a specified number of increases into it and you want to know how many rows it should continue with that many increases, you can also calculate that (formula below).

So for a gauge where a = the stitch width in inches, b = the stitch height in inches, and n = the number of rows in a pattern band,

number of increases in that band = pi * n * b/a

If you’d rather work out how many rows have to go with a particular number of increase stitches m, the formula is:

number of rows = (a*m)/(pi * b)

Very technical, but yes, it makes sense. Now… the problem I came up with was that I COULD make the shawl perfectly round and sneak increases in wherever they fit best and calculate how many non-increase rows went between the increase rows, it would be a bitch to chart and it wouldn’t be easy to remember which rows were increases, and therefore really difficult to knit. It’s possible I would do it for an art piece, but for a pattern that I intend other people to do? I’m just not that sadistic.

So I went with a half of an octagon. Figuring that if you increase 2 sts each row for a triangular shawl (4 sts every other row averages out to 2 sts per row), that means that over 4 rows, you increase 8 sts. So if I divide the shawl into 4 equal wedges, that means that I need to increase 2 sts per wedge every 4 rows for a total of 8.

And that’s where I am. I drew up my charted wedge with one increased stitch on each side every 4 rows, with a plain stitch between each wedge (like the center stitch down the middle of a top-down triangular shawl and a garter stitch border to give non-curling edges.

It’s not going to be the half-round shawl I envisioned, but it certainly won’t be a triangle.

M