Deriving the obvious – Diagram Monkey

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Diagram Monkey

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July 19, 2026

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.

On the left in grey, a flat hexagonal crochet with 6 rows. On the right a sort of beigey crochet with 6 rows that’s much more circular.

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,...