This week’s geek craft is a tutorial and mathematical analysis of a perfect crochet sphere.
So very... round.
A lot of the legwork for this kind of sphere was done by Ms Premise-Conclusion on her blog post: Ideal Crochet Sphere. There she explains her model and compares the standard method to the ideal case. A good read for those who like crochet and math.
If you don’t really care about the math, she also has a wonderful pattern index of a large set of perfect spheres in a PDF you can download from her pattern page.
One thing she fell a little short of here, was explaining the application of her method. Looking at graphs and her work, I’ve pieced together her model and I’ll go over some uses of it. Namely for producing ideal spheres and for piecing together an easier estimation.
Typically, when you want to make a sphere, you start with a circle of 6 stitches and just increase by 6 stitches every row until you’re about as big around as you like. You crochet a handful of rows at this desired number of stitches, and then decrease by 6 stitches every row until you’ve gotten back to 6 stitches. There is mathematical precedent for why 6 is the best number to use here, which I will hit later.
The issue here, is that this produces a fairly rough estimation of a sphere, and the “do a handful of rows in the middle” portion isn’t very exact. Do too many rows and you get something that’s a bit oblong; do too few rows and your sphere will look a bit disc shaped. Luckily there are enough sphere patterns out there that finding something about the size you want is pretty easy, but if you want to do it yourself, you may be in for a little trial and error to get something that looks right. The rule here is that your sphere should have twice as many stitches on the center row as you have total rows. E.G. if you want a sphere that has about 40 stitches around the middle, your sphere should be 20 rows tall. This is due to the geometry of the single crochet being almost twice as long as it is wide when doing rounds (estimation). If you really want to get specific, you could gauge your stitches and then use that ratio to determine your sphere stitch ratio. In practice, though, 2:1 stitches to rows ratio works pretty well. Deviations from this up to 2.5:1 and as low as 1.7:1 also seem to produce acceptable spheres, though the geometry can depend heavily on how tight you crochet, and how firmly you stuff the sphere. There’s also the limitation that your diameter should be directly based on multiple of your base circle. Again, typically 6 because it works out better.
I will warn you all right now, lots of math and figuring lies ahead!
But, in the engineering world, a project isn’t worth doing unless it’s worth overdoing. So we want a perfect, or near perfect, sphere instead of a pretty wide estimation of one.
This is where Premise’s model comes into play. Using her model you can produce BOTH “ideal” estimations and easy estimate sphere patterns… provided you’re willing to do some math, or at least be willing to use my excel equations (tackled in part 2).
When Premise did her model, she noticed that a sphere basically follows a curve that is directly proportional to the sin(θ) as θ goes from 0 to π. This is useful because any crochet sphere can be broken down into the number of rows that make it up. That is to say the number of rounds from the base set of 6 single crochets in your loop to the 6 closing rounds at the top. We shall call this number of rows N.
Since the diameter of the row is directly proportional to the rows placement along the curve of the sphere (I.E. proportional to sin(θ)) we can closely estimate the number of stitches in the row by multiplying the total number of rows by sin(θ). However, we also have to factor in that a single crochet stitch is roughly twice as tall as it is wide when crocheting with a hook small enough to produce a closed pattern in the round*. Thus, any given row that is located at θ will have 2Nsin(θ) stitches in it. However, this isn’t entirely helpful as it is a continuous function, and not immediately useful (unless you want to graph it and eyeball the number of stitches).
So, we need to come up with a discrete function that provides the number of stitches for any given row. This leads to two important factors of the model: 1) There is an invisible, or 0 stitch, row that corresponds to π that needs to be included; 2) Any given row can be easily represented by a multiple of π divided by the total number of crocheted rows, N, plus 1 to represent the invisible row. Thus, our final model becomes:
Now, before you run screaming from the room, let me explain what that equation is doing and show you an easy way to use it in excel. So don’t panic.
Basically, what the function is doing is breaking the sphere into each row (n) and then calculating the number of stitches in the row (xn), rounded to the nearest integer. That funny “bounding” portion is basically just there to say that n can only be a whole number that is between and including the first row, and the last row of the crochet.
Luckily, this can be pumped into excel pretty easily. Open up excel and highlight box C1 and put a 10 in it, this is just a place holder right now.
Select box A1 and type: “=ROUND(2*($C$1+1)*SIN(ROW(1:1)*PI()/($C$1+1)),0)” without the quotes and hit enter. Now highlight B2 and enter: “=A2-A1” without the quotes. Drag both boxes down for a bit to create a two column string of numbers. Any given row of this integer list is the number of stitches in that particular row, and the number of stitches you’ll be adding when you crochet that particular row. You can pump the number of rows you want to have in your sphere into C1, and it will update the values to your new sphere.
You ignore everything after the row you're crocheting to, in this case row 20 is the last row you would crochet.
The big issue here is that you’ll have rows that don’t work out so well. In the basic 10 row sphere, you’ll see that row 3 would have you add 5 stitches to the 12 stitch row below. You could finagle that a bit and do 1 sc, 1 inc, 2 sc, 1 inc, 1 sc, 1 inc, 2 sc, 1 inc, 1 sc, 1 inc. As your sphere gets bigger, though, these kinds of setups become more difficult/annoying to work out.
So, we can look at our model for some estimations by looking at the neighbors of any row. Taking row 3 again, we see that if we do a 4 stitch incrase instead of 5, it simplifies both rows 3 and 4. Instead of doing a 5 stitch increase in a 12 stitch row, and then trying to do 3 in a 17 stitch row, we can instead do 4 stitches on the 12 (thus increasing every 3rd stitch), and then do 4 on 16 (increasing every 4th stitch). That works out quite a bit better, and you’ll still have 20 stitches at the end of row 4. Very nice!
From here, we can begin to develop a more accurate estimate for producing a sphere that is much closer to ideal than the current estimated method. I shall cover this new estimated model in part 2 of the perfect crochet sphere.
-Confusion is a state of mind, or is it?
*The factor for determining the number of stitches per row could be more accurately achieved by using a gauge. For this model, a factor of 2 is used because it’s used to determine the estimated sphere and is relatively close to the actual case. It’s important to note that the curved geometry of working in the round CHANGES this ratio from the row geometry of a single crochet when worked flat which is generally accepted as being about 1:1.