Stacked increases knit a ball with flat sides

A ball knitted in rounds washes the other way built a ball outward from one pole, round after round, increasing five stitches at a time on the rounds where a sphere’s circumference asks for more. It counted the stitches exactly and then drew the ball as a surface of revolution, and it said in its own limitations what that drawing assumed: that the increases were spread round the ball rather than stacked.

A knitter chooses. The five increases of each increase round can go anywhere in it, and the two habits are to shift them along each time, so they spiral, or to put them in the same five places every time, so they stack into five lines from pole to pole. The earlier essay left the second case as a question: stacked, is the ball a sphere or a polyhedron of five faces?

It is neither, and what it is can be computed exactly without knowing anything about the yarn.

Between two lines the knit is flat #

Take any stitch that is not on an increase line. Its neighbours are the stitch before it and after it in its round and the stitch above and below it in its wale — exactly the neighbours it would have in a flat piece of plain knit. The round it sits in is longer than the round below only at the increase lines, so between two lines the rounds are the same length stitch for stitch, and the fabric there is flat knit, with straight rows and straight wales.

So a ball with stacked increases is $m$ flat panels sewn together along $m$ lines. And a closed ball has a total curvature it cannot avoid — Gauss’s theorem fixes it at $4\pi$ for anything shaped like a sphere, whatever its details. Flat panels carry none of it. All of the ball’s curvature sits on its increase lines and at its two poles. That is the whole difference from the spiralled ball, whose increases are points scattered over the surface and whose curvature is spread among them.

Flat panels close up one way #

A flat panel with straight rows can bend, but only about lines along which it stays straight. Close $m$ of them up round an axis without stretching any and there is exactly one way to do it: each round becomes a regular polygon, one panel’s row laid flat across each side.

A row of $w$ stitches is a side $w$ wale spacings long, so the round’s distance from the axis to the middle of a side — its inradius — is

$$\rho = \frac{w}{2\tan(\pi/m)},$$

and its corners, where the increase lines run, stand $1/\cos(\pi/m)$ further out. Down the middle of each panel runs a straight wale, and it has to keep its own length: each course climbs by as much as the course’s length allows once the round’s growth has taken its share. That is the whole construction, and it is an exact isometry — the unstuffed ball, not an approximation of it.

The rows a sphere asks for grow as the sine of the latitude, so the polygon’s inradius grows as $f$ times the sphere’s radius times that sine, with

$$f = \frac{\pi/m}{\tan(\pi/m)},$$

which is 0.865 at five lines and 0.967 at ten. Everything about the faceted ball comes from $f$.

The pole is the disc, stacked #

The pole of a round-knitted ball is a knitted disc, and that essay found the condition for a disc worked in rounds to lie flat: $m$ increases every round grow the circumference by $m k_r$ per course of radius, flat when that is $2\pi$. Five increases every round is flat at an aspect of 1.257, inside the band plain knit moves through as it relaxes, 1.25 to 1.29.

That condition was derived for increases spread round the disc, and stacking changes it. Stacked, each panel is a flat triangle whose half-width grows by $k_r/2$ per unit of its centre wale, and $2m$ half-triangles close flat only if their angles add to a full turn:

$$2m\arctan\frac{k_r}{2} = 2\pi, \qquad k_r = 2\tan\frac{\pi}{m}.$$

Five stacked increases a round would need an aspect of 1.453 to lie flat, far outside anything plain knit reaches. Its triangles fall short of a full turn, and the disc rises into a pentagonal point whose faces meet the axis at 62° in the fully relaxed state — 59° dry, 63° wet. The spread condition, $mk_r = 2\pi$, is the small-angle form of the stacked one; the difference between them is exactly the difference between an arc and its chord, and at five lines it is large.

Each edge is a fashioned edge #

A panel’s edge is where its increases stack, and it steps out by half a stitch at every increase round on each side. That is a fashioned edge, and a fashioned edge has a quantised angle: half a wale every course leaves the wale at $\arctan(k_r/2)$, 32.6° in the fully relaxed state, the same in any yarn at any gauge, because the loop length cancels out of the aspect.

Further from the pole the increase rounds spread out, one every two rounds and then every three, and the edge steepens through the smaller fashioned angles towards the wale, until at the equator it runs straight. The panel’s edge is therefore a sequence of the quantised angles, one for each gap between increase rounds, which is why it is drawn as a curve here and knits as a staircase. The same edge on a ball knitted sideways in short rows, whose turns have to slow down on the same cosine, is laid across the rows rather than along the wales; a gored ball’s panels are not flat, because a short row puts its curvature across the whole panel rather than on its edge.

The equator is a polygon across the circle #

The equator’s round has the circle’s stitches, so the polygon has the circle’s perimeter and sits across it: its flats inside, its corners outside.

At five lines the flats are 13.5 per cent inside the circle and the ridges 6.9 per cent outside, and the round turns through 72° at each ridge. At eight lines the flats are 5.2 per cent in and the round turns through 45° at each; at twelve, 2.3 per cent and 30°. The facets are a quantity that falls off as one over the square of the lines, because a regular polygon approaches its circle that way.

Taller than it is wide #

A polygon’s inradius grows more slowly than a circle’s radius would with the same perimeter, because a straight side is a shorter way round than an arc. So each round of the faceted ball is narrower than the sphere’s at the same count, and the wale down the middle of a panel, keeping its length, has more of that length to spend climbing.

The unstuffed ball is taller than the sphere its counts came from. At five lines it stands 2.42 sphere radii tall, 2.14 across its ridges and 1.73 across its flats: 1.13 times as tall as it is wide across the ridges, and 1.40 across the flats. At ten lines it is 1.06 and 1.11; at twenty, 1.02 and 1.03.

So the knitter’s word for what a stacked pattern makes — a lemon, a pumpkin, a ball with seams — is the right one, and it has a number. The poles are points because near a pole the rounds’ inradius grows at $f$ per wale of meridian rather than at one: the pole is an $m$-sided pyramid whose faces meet the axis at $\arcsin f$, 60° at five lines against the 90° of a flat cap, and 75° at ten.

A round in rounds is its own measure #

It is worth saying where the knitted counts come into this, because they come in twice.

The shapes above are for a smooth sine, which isolates what stacking does. A real ball’s counts are rounded to whole increase steps, and the essay on round-knitted balls found those roundings have their own consequences for its shape. For the forty-round ball of five-stitch steps, stacked, the counts give a height over width of 1.13 across the ridges and 1.40 across the flats, against 1.02 for the same counts with the increases spread round: stacking alone accounts for the difference.

At ten stitches a step the counts themselves already make an elongated ball — 60 stitches at the equator where 62.5 were asked for — and the stacked and spread versions give 1.37 and 1.38. There the rounding is the larger effect and the stacking is invisible beside it.

What rounding it costs #

Stuffing pushes a ball towards round, and it can only do it by stretching knit. The quantity it has to supply is the mismatch between a flat panel and a panel of a sphere.

On a sphere every meridian has the same length, pole to pole: the one down a panel’s middle and the one along its edge. On the flat panel they do not. The centre wale is straight; the edge, where the increases stack, runs out and back as the row widens and narrows, and is longer. For a smooth sine the excess is

$$\frac{1}{\pi}\int_0^{\pi}\sqrt{1 + \left(\frac{\pi}{m}\right)^2\cos^2\theta}\;d\theta - 1 \;\approx\; \frac{\pi^2}{4m^2},$$

9.2 per cent at five lines, 3.8 at eight, 2.4 at ten and 1.0 at sixteen. That is the stretch the stuffing must put into the middle of every panel, lengthwise, or take out of its edges, before the ball can be a sphere. Short of it, the ball stays part-way to its faceted shape.

Set against a wash #

Whether a few per cent of stretch is a lot for plain knit depends on what the knit does anyway, and one thing it does without any force at all is relax. A knit’s dimensions come from its loop, and the constants that turn a loop length into course and wale spacings change from one relaxed state to the next. By Munden’s constants a course’s height changes by 6.0 per cent between the dry and wet relaxed states and by 3.8 per cent between wet and fully relaxed.

So the comparison has a crossover. At eight lines the stretch that rounds the ball equals what a wash moves a course by; at ten it is less. Below that, at five or six lines, rounding the ball asks the stuffing for more lengthwise stretch than the fabric’s whole range of relaxed states covers, and the panels show. The comparison is only a scale, not a mechanics — a knit under stuffing is under load, and what it gives under load is not what it gives in relaxing — but it says which side of ordinary the demand sits.

Staggering the lines quarters the cost #

The arithmetic also says what the spiralling habit buys, at least in its simplest form. Alternate the increases between two sets of five lines — the first increase round on one set, the next on the other — and the ball has ten lines, each panel half as wide, each line increasing half as often. The counts are unchanged and the geometry is the ten-line geometry: the stretch falls from 9.2 per cent to 2.4, a quarter, as $\pi^2/4m^2$ says it must when $m$ doubles.

A true spiral, shifting every increase along by a stitch, spreads the curvature further still, and the surface of revolution the earlier essay drew is its limit. What the stacked case shows is that the choice between stacking and spiralling is not cosmetic: it moves the stretch a ball needs by a factor that can be larger than any relaxation the fabric undergoes.

A hat’s crown is the same pyramid #

A hat’s crown is the top of this ball, and knitters decrease it both ways: in stacked lines, which make a star, or in a swirl. The model says what the star is. Near the pole the stacked rounds are polygons of inradius $f$ times the flat circle’s, so a crown decreased in eight stacked lines is an octagonal pyramid, its faces meeting the axis at 71.5° rather than lying flat, its flats 5.2 per cent inside the circle its stitches would make. A crown decreased in five lines is a pentagonal pyramid with a 60° point.

A sock’s heel turns a corner the same way: a heel turns a right angle because of the loop, its short rows adding length down the back line, and the heel’s two gusset lines are a pair of stacked shaping lines with flat knit between them. The heel is a corner that is meant to be a corner, and it is sharp for the reason a stacked ball’s ridges are.

The same geometry appears in why clothes need darts, from the other side. A dart is a line along which flat cloth is gathered so that it can curve, and every increase line here is a dart run in reverse, adding cloth along a line instead of taking it away. A garment’s darts make it faceted for exactly this reason, and a tailor’s cure is the knitter’s: more of them, smaller.

The model named #

The fabric is plain knit with straight rows and wales between the increase lines, and the unstuffed ball is the unique unstretched way flat panels with straight rows close round an axis: regular $m$-gonal rounds of inradius $w/2\tan(\pi/m)$, each panel’s centre wale keeping its length. The counts are the forty-round ball’s from the earlier essay, rounded to steps of $m$ from $2\pi R\sin$ and smoothed over three steps, and the smooth case takes the sine unrounded so that only $m$ differs from a sphere. The stretch is the length of a panel’s edge over its centre wale, less one. The wash is Munden’s course constant, 5.0, 5.3 and 5.5 in the dry, wet and fully relaxed states.

What was counted #

Twelve increase counts from three to twenty lines on the forty-round ball’s own counts, and twenty-eight from three to thirty on the smooth sine, each with its height, its width across ridges and flats, its pole angle and its stretch; the stretch was required to agree with $\pi^2/4m^2$ to leading order. The fully relaxed state throughout, with the three relaxed states compared for five lines, where the stretch moves between 8.0 and 8.5 per cent.

What the picture cannot show #

Stuffing. Every shape drawn is the unstuffed ball. A stuffed ball is that shape plus a pressure, and it moves towards round by stretching the panels’ middles and gathering their edges; how far it gets depends on the knit’s resistance to exactly that deformation, which nothing here computes.

The lines themselves. An increase is a stitch with an unusual structure, and a line of them is a seam of slightly different fabric — stiffer or looser than plain knit depending on the increase. The model gives the line no width and no properties of its own. In a real ball a line of lifted increases stands a little proud and a line of yarn-overs is a row of holes, and both change how sharp the ridge reads.

A knit’s rows are not perfectly straight. A panel of plain knit can shear a little, which lets its rows curve in the plane and softens the polygon’s corners without any stretch. That is a second way for stuffing to round the ball and it is not in the model; it makes every number here an upper bound on how faceted a real ball is.

Who worked out which part #

The stacked and spiralled crown, and the star one makes, are knitters’ knowledge. That a surface made of flat pieces concentrates its curvature on their seams is Gauss’s, and the gored balloon, the beach ball and the geodesic dome are all engineering on it. What is done here is to find the one unstretched shape a stacked-increase ball can take from its counts, to put the stretch that rounds it at $\pi^2/4m^2$, and to set that against how far plain knit moves between its own relaxed states.

Still open: where on the panel the stuffing stretches #

The stretch is a length difference between a panel’s middle and its edges, and it says nothing about how the stuffing distributes it. A panel could stretch uniformly along its middle, or bulge across, or shear so its rows bow; each gives a different stuffed shape, and each leaves a different residue of the facets.

The deciding quantity is the ratio of a plain knit’s lengthwise stiffness to its shear stiffness at the small loads stuffing applies, and the knit relaxes for as long as it is allowed suggests those loads are ones the fabric keeps creeping under. Whether a stacked ball becomes rounder the longer it has been stuffed — the facets relaxing out over days as the panels creep — is the question that would turn the unstuffed shape into a prediction about the toy on the shelf.