Deriving the obvious

It’s too hot to do anything at the moment other than lie on the sofa with a fan pointed at my face and think. What I was thinking about was crochet and amigurumi and how a simple flat spiral can become such an astonishing array of shapes. You just follow the pattern, making one knot after another and the whole thing just kind of folds itself up into the right shape. Or, if you are me, often, the wrong shape.

Lots of the patterns start in the same way: you make a little ring with six “single crochets” in it, then you go round and crochet two stitches into each of the crochets (known as an increase, presumably because it increases the number of stitches) in the first row, so now you have 12 stitches arranged in a flat circle. Then it gets a bit more complex (and this is where I start to lose count) because you have to alternate single crochets and increases till you’ve gone all the way around once again at which point you have 18 stitches. After that it’s 2 single crochets and an increase repeated over and over which gets us to 24 stitches.

Up till now the spiral of knots is still completely flat and one starts to worry that it will look nothing like the cute fox (or beaver or octopus or monster¹) or whatever that is shown in the pattern². Sometimes the pattern goes on this way for quite a while with the nth row consisting of six lots of n-2 single crochets followed by an increase. I just got done stitching a pattern that followed this rule all the way to 12th row at which point I had a pretty yet completely flat crocheted coaster, and not the jolly yellow sphere that the pictures promised me.

If you forget that you’re stitching in a spiral for a moment, and assume that the stitches are approximately square (with sides δ long), then for a flat circle, the circumference of the nth row is 2πδn and thus you have 2πn stitches in the nth row. Two lots of π is approximately 6 so this matches pretty neatly the typical pattern.

It’s at the point when the pattern deviates from this rule that the magic happens: On the 13th row and all the way up to the 22nd row, the pattern said to make 72 single crochets with no increases. This is the bit I like because you don’t need to keep count, you can just switch off and stitch³. You just need to keep track of where each row starts and ends which I do by keeping an old bit of yarn as a sort of bookmark.

Unfortunately, that frees the brain up for thinking, which is always a mistake.

When you stop with the increases, and just do single crochets, what you end up with is a cylinder, which isn’t a sphere either. Indeed, I’ve made amigurumi which were cylinders that followed this exact same pattern⁴ but ended up looking spherical.

What made them spherical (or approximately so) was the stuffing and the fact that there’s some tension and give in the yarn. Even if you start with a cylinder, once the stuffing goes in, the flat ends start to dome out and the straight sides start to bulge and because the whole thing is somewhat elastic, the doming top and bulging sides pooch out into a pretty good sphere.

This drives me slightly nuts.

I feel like the pattern ought to do more work so that the curvature arises naturally out of the pattern of stitches and not simply because a sphere is the shape that maximises the volume for a given surface area.

It’s not the only thing that bothers me. If you follow the pattern above, you don’t actually get a circle, what you get is more like a hexagon (see the grey crochet in the picture below). If there are six single crochets followed by an increase then the six single crochets prefer to be in a straight line and the increase, because it has two stitches in one, tends to put a bend in the row at that point. Some patterns align all the increases on consecutive rows, but this emphasises the hexagonal shape and we end up with something more like a distorted hexagonal prism.

The benefit of the hexagonal prism is, it’s easy to remember the pattern or work out what you should do in a particular row by counting how many rows you’ve done. It’ pretty simple to mitigate the hexagon effect by offsetting alternate rows (see the pinky beige crochet in the image above)

Crocheting gives you a lot of time to ponder and for quite a lot of that time I have been pondering how to pattern the stitches so that the resulting surfaces are naturally curved. To achieve that you need fewer increases, so that the surface starts to curve, but how many fewer⁵?

If we think of a sphere and assume that each stitch from the first to the last is the same size δ then each stitch makes an angle θ of around δ/r where r is the radius of the sphere. If we want a pattern with R rows then θ=π/R. Approximately, because the stitches are slightly different shapes depending on where they occur and what the neighbourhood looks like.

So the angle from the north pole down to the end of the nth row is nθ and the radius of the row is r sin(nθ) and its circumference is therefore 2πr sin(nθ) and that row will have 2πr sin(nθ)/δ stitches.

We can calculate δ because we know we want R rows and these need to fit in half the circumference of the sphere: δ = rπ/R.

Subbing everything in:

Stitches in row n = 2πr sin(nπ/R)/ (rπ/R) = 2R sin(nπ/R)

If we want 20 rows then the number of stitches in the first ten rows are (with rounded numbers in brackets and the standard pattern in square brackets):

1. 6.257378602 = (6) [6]
2. 12.36067977 = (12) [12]
3. 18.15961999 = (18) [18]
4. 23.511410092 = (24) [24]
5. 28.28427125 = (28) [30]
6. 32.36067977 = (32) [36]
7. 35.64026097 = (36) [42]
8. 38.04226065 = (38) [42]
9. 39.50753362 = (40) [42]
10. 40 = (40) [42]

The next ten rows are the same but in reverse order with zero stitches at the end of the 20th row because at that point we have reached the south pole.

It’s kind of nice that the first four rows come out quite close to the standard pattern⁶, but it already deviates at the fifth row with two less stitches which starts to give us the curvature. The sixth row has four fewer stitches and the seventh row, six fewer. After that I had the standard pattern stay at 42 and the more spherical numbers start to catch up (although they never quite get there).

The pattern is only slightly more complex. In regular notation⁷, the standard pattern is nice and simple:

1. 6sc in a magic ring [6]
2. 6inc [12]
3. 6(sc, inc) [18]
4. 6(2sc, inc) [24]
5. 6(3sc, inc) [30]
6. etc etc [6n]

My pattern is slightly less user friendly, particularly as one gets towards the equator of the sphere on the pole to pole trek. To show the pattern in the pattern, I haven’t staggered the increases to avoid it becoming hexagonal but that’s an easy modification⁸:

1. 6sc in a magic ring [6]
2. 6inc [12]
3. 6(sc, inc) [18]
4. 6(2sc, inc) [24]
5. 4(5sc, inc) [28]
6. 4(6sc, inc) [32]
7. 4(7sc, inc) [36]
8. 2(17sc, inc) [38]
9. 2(18sc, inc) [40]
10. 40sc [40]

After that you have to do the whole thing in reverse but substituting “decreases”, which combine two stitches into one, for increases.

Crocheting it is straightforward but requires a tiny bit more concentration (or more recounts of stitches) than the simpler plan once you get to row 8. Satisfyingly, the spherical curvature becomes apparent already in the 5th row. Annoyingly, it looks slightly too “pointy⁹“, but stuffing sorts that out.

It’s still not quite right: each crochet based on a magic circle is a kind of big long not-quite spiral and not not-quite concentric rings. The stitches aren’t quite square and can vary in size and shape depending where in the process they appear and how tightly you crochet. Still, it’s a slight improvement – in some indefinable sense – and I now have a spreadsheet so that I can make spheres of different sizes.

None of this is revelatory in any sense and certainly nothing new. Looking through the patterns I have, some use the sphere pattern I derived (or part of it or something close to it), others don’t. I just like knowing why things are the way they are.

-fin-

1. Recently, there has been much discussion of the El Nino that is brewing in the tropical Pacific and the forecasts of its possibly stupendous magnitude. People have struggled to find appropriate vocabulary. El Nino events are most often classified, somewhat drily, as weak, moderate, or strong, but definitions and thresholds vary. The forecasts suggest something might be necessary beyond this simple scheme. The word “super” has been used – super El Nino – but it has unfortunately positive connotations: this El Nino will be just super. Others have groped around for an appropriate metaphor and happened upon beasts both real (a 800 pound gorilla) and mythical (Godzilla – brilliantly in this extract from the Houses of Parliament website). A 800 pound gorilla has, from a scientific standpoint, a pleasing specificity even if it is expressed imperially. Godzilla is another matter. Fictional Godzillae vary dramatically in size. I know this because one thing the internet has in astonishing plenitude is images comparing the sizes of monsters. All are horrifyingly big, but some are more horrifyingly big than others and some are frankly ridiculous. When I was younger and dinosaurs of increasingly staggering sizes were being unearthed, I developed the notion that there was no actual upper limit on how big they might become, and ignorance of this important matter was maintained by our mistaking the bones of the larger specimens for mountain ranges and other geographical features. Anyway, Godzilla is clearly an unscientific term for an El Nino. All of which considerations were going through my head during an interview on Irish morning radio and there is an alternative universe – if certain theories are correct, an infinity of them – in which these images gained the upper hand over strict decorum and I entered into a long discourse about El Nino involving an 800 pound gorilla and a water bed while a producer somewhere in Dublin was busy pulling agitated faces and making throat slashing gestures at the bewildered interviewer. ↩︎
2. Sometimes, of course, the pattern is wrong. I got a crochet kit at the kind of shop that has huge boxes full of random stuff that wouldn’t sell elsewhere and I suspect that the pattern was written by an AI. It was complete nonsense. Ho hum. AI: ruining even the simple pleasures in life. ↩︎
3. or knot or whatever… I do not know the lingo. ↩︎
4. OK, generally with one minor modification that the point at which you transfer from the top to the sides, the stitches aren’t regular crochets, but only go through the back loops of the stitches. This puts a break in the surface where the tension along the rows is maintained, but the tension between rows is reduced. This gives a sharper edge than you would otherwise get. ↩︎
5. If you go the other way and add more increases, you get crazy frilly patterns but that way lies hyperbolic geometry. ↩︎
6. The math above can be handy for improvising too. If you want a flat disk starting at any point, then all you need to do is take however many stitches there currently are and add 2π stitches – so 6 basically – to work out how many should be in the next row. It’s then just a matter of spreading the 6 increases out evenly along the row. ↩︎
7. There are several regular notations. In this case, sc is single crochet and inc is increase. A magic ring is grrrrrr. Different notations in different languages don’t necessarily map neatly onto each other which I learned the hardway by having to unpick several hundred stitches I just spent several hours sitching. ↩︎
8. Just start every other row half way through the first lot of sc’s and make up the one’s you missed at the end of the row. ↩︎
9. I suspect that this is cause by the fact that the number of stitches in the first and second rows have to be rounded down from slightly less than 12.5, and for larger spheres (above 100 rows) the second row has slightly more than 12.5 stitches, so 12 gives it a slightly pinched quality, but the effect is small and the stretch in the yarn covers it and I can’t entirely discount my cruddy crocheting skills as the cause. ↩︎