A ball knitted in rounds washes the other way

A knitted ball’s short rows have to slow down worked the sphere sideways, in gores: short rows turning one stitch apart at each pole and further apart towards the equator, on a schedule set by a sine. It ended by naming the other way to knit a ball, the way a hat’s crown is extended past its widest round and closed again — outward from one pole in rounds, increasing and then decreasing.

That construction has to follow the same sine, and it does it with increases instead of turns. What it does not share with the gored ball is where the loop’s aspect sits in its arithmetic, and that single difference decides how each ball changes when it is washed. They change in opposite directions.

The same sine, counted the other way #

A round is a ring. Knitted outward from a pole, round $i$ sits $i + \tfrac12$ course spacings down the meridian and holds $S_i$ stitches, so its circumference is $S_i$ wale spacings. A sphere asks for a circumference of $2\pi R\sin(s/R)$ at meridian distance $s$, and with $N$ rounds from pole to pole the meridian is $N$ course spacings long. Measured in wale spacings, that makes the sphere’s radius

$$R = \frac{N}{\pi k_r},$$

where $k_r$ is the loop’s aspect, how much wider a wale is than a course is tall. The aspect is underneath here. In the gored ball the meridian was counted in stitches — wale spacings — and the circumference in rows, so the aspect multiplied: $K = Nk_r/m$ row pairs at the equator. Here the meridian is rows and the circumference stitches, so it divides: the equator holds $2N/k_r$ stitches.

At forty rounds that is 62.5 stitches in the fully relaxed state. Increases come $m$ at a time — one at each of $m$ points round the ball — so the equator’s count has to be a multiple of $m$: 65 at five a time, 60 at ten.

The increase rounds are where the rounded sine steps. Five at a time, the forty-round ball increases on rounds 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 14 and 16 from the pole: every round at first, then with a gap, then every other round, then every second or third. Ten at a time it increases on rounds 3, 5, 8, 11 and 15. It is the gored ball’s slowing turns in a different currency, and it slows by the same factor, one over the cosine of the latitude, because the sine it is following is the same sine.

The pole is the disc worked in rounds #

Near the pole the sine is straight, so the pole of a round-knitted ball is a flat disc worked in rounds, and the disc essay already has its arithmetic: $m$ increases every $k$ rounds grow the circumference by $m k_r/k$ per unit of radius, flat when that is 2π.

Five increases every round is flat at an aspect of $2\pi/5$ = 1.2566, and so is ten every two rounds — the same growth, laid down half as often in twice the amount. Plain knit’s relaxed aspect is 1.2500 dry, 1.2927 wet and 1.2791 fully relaxed. So the round-knitted pole, like the gored one, is within a couple of per cent of flat in every state, and it crosses flat inside the band.

The difference is the sign. The gored pole’s growth per unit of meridian is $m/(\pi k_r)$ of flat — it falls as the aspect rises, because the aspect is underneath. The round pole’s is $mk_r/2\pi$ — it rises. Every other difference between the two balls is a consequence of that one.

The increases slow on the same cosine #

Away from the pole a sphere’s circumference grows more slowly, at $2\pi\cos\lambda$ per unit of meridian where $\lambda$ is the angle from the pole. A round-knitted ball meets that by spacing its increase rounds further apart: $m$ stitches are added every

$$\Delta i = \frac{m\,k_r}{2\pi}\cdot\frac{1}{\cos\lambda}$$

rounds. Five at a time in the fully relaxed state, that is one increase round every 1.02 rounds at the pole, every 1.18 at thirty degrees, every 1.44 at forty-five and every 2.04 at sixty — and then, in the last fifteen degrees before the equator, three, five, ten rounds apart, until the widest band is knitted even.

It is the gored ball’s schedule term for term, with one factor turned over. The gore turned $m/(\pi k_r)$ stitches a pair at the pole and $1/\cos\lambda$ times that further out; the round ball increases every $mk_r/2\pi$ rounds at the pole and $1/\cos\lambda$ times that further out. The cosine is the sphere’s and belongs to neither construction. The aspect is the fabric’s, and which side of the fraction it lands on is decided by nothing more than which of the two directions — along the meridian or round it — the knitting counts in courses.

A hat’s crown is a band thirty-five degrees from a pole #

The same formula reads backwards as a statement about hats. The disc essay found that the commonest instruction for a flat circle — eight increases every other round — makes a cone that rises 58 per cent of its radius, and that the same rate run as decreases is how a hat’s crown is closed.

On a sphere that rate is a latitude. Eight every two rounds grows $8k_r/2$ = 5.12 per unit of meridian fully relaxed, which is 0.814 of 2π, and a sphere grows at that rate where $\cos\lambda = 0.814$: 35.5 degrees from its pole. A crown closed at a constant eight every other round is the cone tangent to a sphere at that latitude — a cap that meets its head’s curvature at one ring and is too pointed above it. A crown that follows the ball’s schedule instead, decreasing every third round, then every other, then every round in the last few, is a spherical cap, and it is the same instruction as the top of a round-knitted ball.

That is why a garment needs darts in a knitted form: a head is curved in two directions, and a cone — curved in one — can match it only along a single circle. The crown’s decreases are a continuous dart, and their spacing is the curvature they supply.

A wash moves them in opposite directions #

A ball’s counts are fixed when it is knitted. What a wash changes is the loop’s shape, and a knit relaxes for as long as it is allowed: dry, the aspect is 1.2500; soaked, 1.2927; soaked and tumbled, 1.2791.

The gored ball’s pole is 1.019 of flat dry, 0.985 wet and 0.995 fully relaxed. Knitted and rested, it has more circumference than a pole can hold and ruffles; soaked, it closes to a very slight point; tumbled, it is within half a per cent of flat.

The round-knitted ball’s pole is 0.995 dry, 1.029 wet and 1.018 fully relaxed. Knitted and rested, it is the one that is nearly flat; soaked, it ruffles; tumbled, it still ruffles, by a little under two per cent.

So the same wash that tidies a gored ball’s poles frills a round-knitted ball’s. Both balls are the same sphere asked of the same fabric, and a knitter holding the two side by side off the needles would see the gored one frilled and the round one neat, and after a wash the reverse.

What that does to the whole ball #

The pole is where the disc lives, and the rest of the ball follows the sine. So the wash’s effect on the ball’s overall shape is smaller than on its poles — but it is not nothing, and the direction is again opposite.

The round-knitted ball is 1.09 as high as it is wide dry-relaxed, 1.01 wet and 1.02 fully relaxed. The wash lowers it and widens it together, because a rising aspect makes every course shorter relative to every stitch: the meridian, counted in courses, shrinks, and the circumference, counted in stitches, does not. The gored ball does the reverse — its meridian is counted in stitches and its circumference in courses, so the same wash narrows its circumference — and the essay before this one found it at 0.83 dry and 0.96 wet.

The band that ruffles is not at the pole itself. It is two or three rounds out, where the ball increases on three consecutive rounds — five a round, which is 1.018 of flat fully relaxed — before the first gap in the schedule. The cast-on ring at the very pole, five stitches round, sits under that band as a slight point.

Side by side, one fabric #

Set the two constructions beside each other at the same meridian length in the fully relaxed state, and neither is a better sphere than the other; they are wrong in different places.

The gored ball is 0.91 as high as wide and its failing is flat caps, where whole-stitch turns cannot resolve the slow height gain a sphere has near its pole. The round ball is 1.02 and its failing is at the same latitude but the other way — too much circumference in a narrow band, because a run of consecutive increase rounds is the disc’s ruffling construction.

The two holes differ too. A gored ball’s pole is a ring of $m$ course spacings, half a stitch in radius for four gores. A round ball’s pole is its cast-on: $m$ stitches drawn into a ring, $m/2\pi$ wale spacings in radius, 0.8 of a stitch for five. The round ball’s gather is larger at the same size and a knitter closes it the same way, with a thread through the cast-on loops.

Sizes that come out round #

The equator’s count has to be a whole number of increase steps, $2N/(k_r m)$ of them, and rounding it moves the ball’s proportions exactly as rounding a gored ball’s row pairs did.

It saws with a period of about $mk_r/2$ rounds, three at five a step, and like the gored ball its amplitude falls as the ball grows. Unlike the gored ball, it is biased: below forty rounds it is taller than wide at every size, 1.04 to 1.43. The reason is at the poles. A round-knitted ball has to start with a cast-on ring of at least $m$ stitches where a sphere asks for almost none, so its first rounds are already wider than the sine and its meridian spends them climbing rather than spreading. The fewer the rounds, the larger that fixed ring is against the ball.

The sizes nearest round are the ones where the equator’s increase steps come out whole — 48, 51, 64, 67, 80 and 83 rounds in the fully relaxed state — and, as with the gores, adding rounds without recomputing the equator’s count moves the ball along the saw.

Which ball for a toy that will be washed #

A stuffed ball forgives one kind of error and not the other. Stuffing pushes outward, and nothing elastic opposes a knit’s first stretch, so a ball whose counts ask for too little circumference anywhere — a slight point at a pole, a flat cap — stuffs round for almost no force. A ball whose counts ask for too much cannot be stuffed smaller, and a ruffle stays a ruffle: a knit is soft because it bends, and compressing its circumference buckles it rather than shortening it.

So the question for a ball that will be washed is which construction has its ruffle on the side the wash leaves it on.

• Gored, the ruffle is at the dry pole and the wash removes it: the finished, washed ball has poles within one or two per cent of flat, on the forgiving side.
• In rounds, the dry ball is the neat one and the wash puts a ruffle into it, a few rounds from each pole, that stuffing cannot remove.

For a toy that will be washed, knit the gores; for a ball that will only ever be dry — a pincushion, an ornament — knit it in rounds. The arithmetic does not care which looks neater on the needles, and on the needles the answer is the opposite one.

How the ball was built #

A round’s count is the sine rounded to a multiple of $m$: $S_i = m\cdot\mathrm{round}\!\left(S_\text{eq}\sin(\pi(i + \tfrac12)/N)/m\right)$, never fewer than $m$, with the equator’s count $S_\text{eq}$ itself the nearest multiple of $m$ to $2N/k_r$ in the state the ball is planned for, and held fixed when the state changes. The increase rounds are the rounds where the count steps.

The surface is built as the gored ball’s was. The edge runs from the cast-on ring through the middle of each step in the count, is smoothed over three steps because a fabric reads alternating increase spacings as their mean, and becomes a surface of revolution whose radius is the count over 2π and whose height climbs by $\sqrt{1 - (dr/ds)^2}$ along the meridian. A piece that grows faster than its meridian is long has no surface of revolution; it is counted and drawn as a ruffle.

The pole’s growth is not computed from the surface at all. It is the disc’s formula, $mk_r/k$ against 2π, and it is the number the washing comparison is made on, so that comparison does not depend on the smoothing.

The loop constants are Munden’s, as for the disc and the gores, and the aspect is their quotient — the one number a knit’s dimensions come from its loop supplies and two knits with one tightness factor found to be one curve for every yarn; the loop length cancels from everything, as it did from the tube, the heel and both of the other shapes.

What the silhouettes leave out #

Where the increases go round the ball is not in them. Five increases a round can stack in five straight lines, which makes a ball of five flat-sided panels, or spiral, which makes it rounder; the surface of revolution is the average and neither extreme. The ruffle’s shape is not computed, for the reason the disc gave. And the stuffed shape is not the natural one: every silhouette is the shape the counts ask for with no force on the fabric, and a stuffed ball is the counts plus a pressure the fabric will yield to in one direction and not the other.

Who worked out which part #

The round-knitted ball is a hat’s crown continued, and knitters have made it that way for as long as they have made crowns; the rule of thumb — increase often, then less often, then knit even round the middle — is the rounded sine as a habit, as the gored ball’s turning rule was.

The washing crossover is the disc essay’s, carried to both poles of a sphere. What this essay adds is that the two ways of making one ball put the loop’s aspect on opposite sides of the same fraction, so one wash moves their poles through flat in opposite directions; and that the choice between them, for a washed toy, follows from which of the two errors stuffing can repair.

Still open: whether stacked increases make a polyhedron #

The silhouette treats a round ball as a surface of revolution, which is the right model for increases that spiral and the wrong one for increases stacked in lines. Stacked, each line of increases is a seam-like ridge along a meridian and the fabric between two of them is a gore — a curved panel knitted in rounds rather than in short rows.

Stacked increases are therefore a gored ball in disguise, with $m$ gores whose widths grow by one stitch at each increase round, and whether such a ball is closer to a sphere or to a polyhedron of $m$ faces depends on how much each panel bends between its ridges. That is the calculation this account has not done: the curvature a panel of plain knit takes between two lines where its width steps, which would say at what number of increase points a stacked round ball stops showing its facets.