A knitted ball's short rows have to slow down

A knitted disc is flat at one shape of loop turned a claim about a surface into one number. A disc is flat when its circumference grows by exactly 2π per unit of radius; a knitted disc’s growth is a count of loops in one direction over a count in the other; so four wedges of short rows lie flat at a loop aspect of 4/π and plain knit’s aspect is within half a per cent of it.

It ended on the obvious next surface. A ball, a toe and a hat’s full crown are not flat — they are curved everywhere by the same amount — and the disc’s single number cannot describe them. A sphere needs a growth that changes, and the knitted question is whether a whole-stitch construction can make one change the right way.

It can, and what it has to do is specific. A ball knitted sideways in gores has its short rows turn one stitch apart near each pole and further and further apart towards the equator, on a schedule set by a sine. Turned at an even spacing, the same gores knit something with no curvature at all.

What a sphere asks of a gore #

A ball knitted sideways is made of gores, like the paper gores of a globe: panels running from one pole to the other, widest at the equator, sewn or knitted edge to edge. Each gore is worked in short rows along its meridian. A row pair runs from one end towards the other, turns, and comes back, and each pair stops a little short of the last, so the rows pile up in the middle of the gore and thin out towards its ends.

So the gore’s width at a stitch is the number of rows passing through that stitch, times the course spacing. Put $m$ gores side by side and the ball’s circumference at a stitch $s$ from the pole is $m$ times that.

A sphere has a circumference at meridian distance $s$ of $2\pi R\sin(s/R)$, and its meridian from pole to pole is $\pi R$ long. Measure the meridian in stitches, $N$ of them, and the rows each gore must carry are

$$W(s) = 2K\sin\!\left(\frac{\pi s}{N}\right), \qquad K = \frac{N\,k_r}{m},$$

where $K$ is the number of row pairs through the equator and $k_r$ is the loop’s aspect — how much wider a wale is than a course is tall, the constant the heel and the disc both turned on. The loop length itself has cancelled, as it does from every shape on this account — it left a tube’s taper the same way — so a ball in a fine yarn and a ball in a coarse one are the same ball at different sizes. What decides the shape is the aspect alone, and a knit’s dimensions come from its loop is where that aspect is measured: Munden’s course and wale constants, whose quotient it is.

The gore’s edge is where its short rows turn, so a schedule of turns is a drawing of $W$. Pair $k$ has to turn at the stitch where the sine reaches $2k - 1$ rows — halfway up its own riser — which is

$$a_k = \frac{N}{\pi}\arcsin\!\left(\frac{k - \tfrac12}{K}\right)$$

stitches from the pole, rounded to a whole stitch.

Turned evenly, a gore makes a cushion #

The disc’s construction turned one stitch every pair, and the tempting way to knit a ball is to keep doing that until the middle of the gore and then come back. That is not a ball.

Turning one stitch a pair adds two rows per stitch at every stitch, so the circumference grows at the same rate all the way to the equator. That is the disc’s growth, and four gores of it are the disc’s four wedges: 2π per unit of meridian, which is flat. The evenly turned ball is two flat discs whose rims meet at the equator.

Its height is 0.17 of its width — five wale spacings high and thirty across, for a meridian that should have made a ball about nineteen across and nineteen high. In the flat-lay drawing it is the diamond behind the sine gore: fifteen row pairs against ten, and one and a half times as wide at the equator. Anything stuffed into it pushes it towards a lens, and the disc essay’s point about asymmetry applies: a knit extends readily and resists compression, so the cushion can be stretched towards a ball far more easily than an over-full gore could be pulled in.

The sine gore asks for a ball, 0.91 as high as it is wide at thirty stitches, with its equator where a sphere’s is and its poles a little flat — for a reason taken up below.

The short rows have to slow down #

Read the sine schedule as spacings and it says the thing a knitter needs. Near the pole the sine is a straight line, so the turns are evenly spaced there; towards the equator it flattens, so the turns spread out.

For four gores at thirty stitches the ten turns fall at stitches 0, 1, 2, 3, 4, 5, 6, 8, 9 and 11 from each pole: seven turns one stitch apart, then twos and ones alternating, and the last pair stopping four stitches short of the middle. The ideal spacing between pairs $k$ and $k+1$ is the difference of two arcsines, and near the pole it is

$$\Delta a \approx \frac{m}{\pi k_r}\cdot\frac{1}{\cos\lambda},$$

with $\lambda$ the latitude measured from the pole. The first factor is the spacing at the pole; the second is what makes a ball of it. The cosine is why the slowing is gentle for most of the gore and abrupt at the end: 1.00 stitches a pair at the pole, 1.15 at thirty degrees, 1.41 at forty-five, 2.0 at sixty.

Rounding decides where the whole-stitch gaps change, and it does so by carrying its own error forward — each turn is placed at the rounded position of the ideal, not at the previous turn plus a rounded gap — so a spacing of 1.3 comes out as a run of ones with a two every third or fourth pair. That is exactly what a knitter does by hand when a pattern says “turn one stitch further in, and every third pair turn two”, and the schedule above is the arithmetic that says where the twos go.

It is the same move a fashioned edge makes in the other direction. A fashioning machine can only narrow a panel by whole wales every whole number of courses, so the angles it can cut are a quantised set, and an angle between two of them is made by alternating. A gore’s turns are a fashioned edge that changes its angle as it goes — steep at the pole, shallow at the equator — and every one of its local angles is one of the fashioned edge’s rational steps.

The pole is a disc, so gores come in fours #

At the pole the sine is a straight line, and a straight-line growth is a disc. So the pole of any ball knitted sideways is the knitted disc of the essay before it, and the disc’s condition applies there unchanged: $m$ gores turning $t$ stitches a pair close $2m/(t\,k_r)$ radians round the pole, and the pole is flat — as the pole of a sphere must be — only when that is 2π.

Solve for the turn at the pole and it is $m/(\pi k_r)$ stitches a pair.

Four gores turn 0.995 stitches a pair at the pole, eight turn 1.991 and twelve 2.986 — whole numbers to within half a per cent, because $\pi k_r$ is 4.02 and the disc essay’s four wedges are its reason. Six gores need 1.49 stitches a pair, which a knitter meets by alternating ones and twos from the very first pair; five need 1.24; three need 0.75, which is less than one stitch a pair and cannot be knitted as short rows at all — a pair would have to turn at the same stitch as the pair before it, which is a dart, and the ball would carry a pleat at each pole. That is not a figure of speech: a dart is what a flat cloth needs to become curved, a wedge of material taken out so the rest can close round a point, and a three-gore pole has too little circumference to be flat and must lose its extra in exactly that way.

So the ordinary knitted ball of six gores is not wrong. It is a ball whose pole has to be made from alternating turns, and whose pole therefore reads as the average of two constructions rather than as either; a four-gore or eight-gore ball has a pole that is one construction.

The pole has a hole, and the ball has flat caps #

Two things about the pole do not come out of the disc, and both are visible on any knitted ball.

The pole is a ring, not a point. The first row pair of every gore passes through the pole stitch, so $m$ gores meeting there leave a circumference of $m$ course spacings — a hole of radius $m/(2\pi k_r)$ wale spacings: half a stitch for four gores, a whole stitch for eight. Every short-row ball draws that ring closed with a thread, and the more gores it has the bigger the gather.

And the caps are flatter than a sphere’s. Near the pole a sphere’s radius grows at $\cos(s/R)$ per unit of meridian — nearly one — and its height grows at $\sin(s/R)$, which is nearly nothing. The height is the sensitive quantity there: a growth of 0.995 per stitch gives a rise of 0.10, and a growth of 0.95 gives 0.31. Whole-stitch turns cannot resolve the difference between 0.995 and 0.95, so the first several stitches of a four-gore ball are knitted at the disc’s growth and rise at the disc’s rate. The silhouette above shows it: flattened caps a few stitches across, and the ball making up the height further down the meridian.

Only some sizes come out round #

The number of row pairs through the equator is $N k_r/m$, and a pair is a pair: $K$ has to be whole. At thirty stitches and four gores it is 9.59, and rounding it to ten puts four per cent too much circumference at the equator, which is why the thirty-stitch ball comes out at 0.91 rather than one.

The roundness saws with the size, with a period of about three stitches — the meridian length that adds one row pair at the equator, $m/k_r$ — and an amplitude that falls as the ball grows: up to 46 per cent off round between twelve and thirty stitches, 12 per cent between thirty and sixty, 9 per cent beyond. The sizes where $Nk_r/4$ is nearly whole — 25, 28, 31, 50, 53 and so on in the fully relaxed state — come out closest to round.

That makes the obvious pattern-book instruction for a size change a trap. Adding stitches to the meridian without recomputing the equator’s row pairs moves the ball along this saw, and one or two stitches can move it from round to a ten per cent egg.

A wash moves the pole #

Plain knit’s loop aspect moves as it relaxes — a knit relaxes for as long as it is allowed — from 1.2500 dry-relaxed to 1.2927 wet and 1.2791 fully relaxed. A ball’s counts are fixed when it is knitted, so the aspect moves the shape.

Dry-relaxed, the four-gore pole grows at 1.019 per stitch of meridian, which is more than a surface of revolution can take: it is the disc essay’s four wedges ruffling in the dry state, now at both ends of a ball. Soaked, the pole’s growth falls to 0.985 and the ball is at its roundest, 0.96 high for its width; tumbled, 0.995 and 0.91.

So the washing crossover the disc found carries straight into the ball, with the same sign. A four-gore ball knitted and rested has two frilled poles; washed, they close. An eight-gore ball does the same at twice the turn, because its pole is two-stitch turns of the same disc, and the ratio $m/(\pi k_r)$ crosses its whole number in the same place.

How the ball was built #

A gore is a list of turns. Pair $k$ is worked across stitches $a_k$ to $N - 1 - a_k$, so the rows through a stitch are twice the number of pairs that reach it; the schedule on the sine rounds each $a_k$ from its arcsine, and the even schedule is $a_k = k - 1$.

The surface is built from the gore’s edge. The edge runs through the middle of each riser — pair $k$’s edge at $2k - 1$ rows at its turning stitch — and is smoothed over three turns, because a fabric does not keep a one-stitch staircase and a schedule alternating gaps of one and two is read by the fabric as their mean. The circumference at each point is $m$ times the rows times the course spacing, $1/k_r$ wale spacings; the radius is that over 2π; and the height climbs by $\sqrt{1 - (dr/ds)^2}$ along each piece of meridian. A piece whose radius grows faster than the meridian is long has no surface of revolution, and it is counted as excess rather than drawn as a shape.

What is required of it. A later pair never turns nearer the pole than an earlier one; the loop constants are Munden’s, as for the disc; and the construction refuses a ball of fewer than two gores or a meridian of fewer than six stitches, because neither has a pole to speak of.

What the silhouette cannot show #

It is the shape the counts ask for, not the shape a stuffed ball takes. A knit is extensible and stuffing pushes outward everywhere, so a real ball is rounder than its counts wherever they ask for less volume than a sphere — the flat caps and the cushion’s rim both — and cannot be made smaller where they ask for more. The silhouette is the natural shape, in the sense the disc’s cone was.

How much rounder is a question about force, and this account’s answer to it is lopsided in a useful way. What stops a knit extending found that nothing elastic opposes a knit’s first stretch — a knit is soft because it bends, and extending a loop bends nothing further — so a knitted ball stuffed firmly will round out its flat caps for almost no force at all. It will not shrink an over-full equator, because compressing a knit buckles it. So the error to prefer is the one on the flat side: a ball whose equator’s row pairs have been rounded down stuffs round, and one rounded up stuffs into a slightly squashed ball, wider than it is tall.

The smoothing is a choice. Three turns is about the distance over which a knitted edge visibly straightens a staircase; a wider window would round the caps a little and a narrower one would flatten them further. Every proportion above moves by a few per cent if it is changed, and none of the counting results — the turn schedule, the gores in fours, the whole row pairs — depends on it at all.

And the ruffle’s shape is not computed, for the reason the disc gave: a surface that grows too fast has no surface of revolution to take, and how it buckles depends on the fabric’s stiffness against its extensibility.

Who found which part #

The globe’s gore is old: printed globes have been made from gores since the early sixteenth century, and an ideal gore’s outline is the same sine for the same reason. Knitters found the short-row ball by hand, and the working rule they settled on — turn one stitch further each pair, then two, then more towards the middle — is the rounded sine written as a habit.

What is new here is that the rule has numbers in it. The spacing at the pole is $m/(\pi k_r)$, which is the disc essay’s four wedges and says why gores come in fours; the slowing is $1/\cos$ of the latitude; and the equator’s row pairs, rounded, decide whether a given size is round. The loop constants are Munden’s and the loop length cancels, as it did from the tube, the heel and the disc.

Still open: a ball knitted in rounds #

A ball can also be knitted outward from one pole in rounds, increasing and then decreasing, the way a hat’s crown is extended into a sphere. Its growth is the disc-in-rounds construction — $m$ increases every $k$ rounds grow $mk_r/k$ per course — and a sphere needs that growth to fall from 2π at the pole to nought at the equator, as $2\pi\cos(s/R)$.

So a round-knitted ball’s increases have to thin out on the same cosine, but with the aspect on top rather than underneath: where the gored ball’s turns spread by $1/\cos$, a round ball’s increase rounds must spread by $1/\cos$ too, and its pole is ten increases every two rounds rather than four wedges. The disc essay found those two constructions crossing flat in opposite directions as a knit is washed. Whether a round-knitted ball and a gored ball therefore change shape in opposite senses on a wash — one pole closing as the other opens — is one more calculation, and it would say which of the two constructions a knitter should choose for a toy that is going to be washed.