A knitted disc is flat at one shape of loop

A flat disc is a demanding object. Its circumference at every radius has to be exactly 2π times that radius — not about that, exactly — because a disc that grows a little less closes up into a cone and one that grows a little more has nowhere in the plane to put the extra and throws it out as a wave round its edge.

Woven cloth never has to meet that condition, because nobody weaves a disc; they cut one. Knitting is different. A beret, a coaster, a hat’s crown and the medallion at the centre of a shawl are all knitted round, and every one of them is a claim that a count of loops can meet a transcendental number.

It can, at one shape of loop. A heel found that the loop’s aspect — how much wider a wale is than a course is tall — is the constant that converts courses into turns. A disc is the same conversion asked of a whole surface instead of one line, and the answer is sharper.

Growth is what a surface is made of #

A flat surface and a curved one differ in one measurable way: how fast the circle round a point grows as it gets bigger. On a plane it grows as 2πr. On a cone it grows more slowly, and on a saddle — or a lettuce leaf, or a ruffled collar — it grows faster.

That makes a knitted disc’s flatness a single number to compute rather than a shape to guess at. Call Θ the circumference the disc grows per unit of radius. Flat is Θ = 2π. Below it the disc rises into a cone whose half-angle is the arcsine of Θ/2π; above it there is no cone that fits and the edge has to leave the plane.

The ratio Θ/2π is the whole measurement, and a knitted disc’s Θ is set by the way it is worked.

Sideways, in wedges #

The first way to knit a disc is to work it sideways. Each course runs out along a radius, from the centre to the edge and back, and the disc grows round the centre as a set of wedges, each made of short rows: a row pair that goes to the edge and turns one stitch short of the centre, then one that turns two stitches short, and so on until the wedge is complete.

A column r stitches out from the centre is worked in 2r of those rows, so its circumference is 2r course spacings at a radius of r wale spacings, and the wedge’s angle is

$$\frac{2}{k_r}~\text{radians}$$

— the same at every radius, which is why the construction makes a true wedge rather than a fan. The loop’s aspect $k_r$ is 1.2791 in the fully relaxed state, so one wedge is 89.59 degrees. Four wedges close 358.4 degrees.

Three wedges would be a steep cone; five would overshoot by nearly a quarter of a turn. Four is the only whole number near flat, and it is near flat because $2\pi k_r/2$ happens to be 4.02.

Four wedges are flat at 4/π, and plain knit is almost exactly that #

Set four wedges equal to a full turn and solve for the aspect:

$$4 \cdot \frac{2}{k_r} = 2\pi \quad\Rightarrow\quad k_r = \frac{4}{\pi} = 1.2732.$$

That is the aspect at which a four-wedge disc lies flat, and plain knit’s own aspect in the fully relaxed state is 1.2791 — within half a per cent of it. The four wedges close 358.4 degrees and the disc is short by 1.6.

That sounds like nothing and it is not. A cone’s height rises as the square root of its angle deficit, not in proportion to it, so a disc short by half a per cent of its circumference rises by 9.5 per cent of its radius. A four-wedge coaster twenty centimetres across stands nearly a centimetre proud at its centre in the fully relaxed state. The smallest misfits are the ones a disc shows most, relative to their size, because the first sliver of deficit buys the most height.

Outward, in rounds #

The second way is to start at the centre with a few stitches and knit outward in rounds, adding stitches at m evenly spaced points every k rounds.

Now each round is a ring and the circumference grows by m wale spacings for every k course spacings of radius:

$$\Theta = \frac{m\,k_r}{k}.$$

The aspect is on top this time. In the wedge construction it was underneath, because there the radius was counted in wales and the circumference in courses; here it is the other way round. So the two constructions answer the same number in opposite senses: a wider loop makes a wedge disc grow less and a round disc grow more.

Ten increases every two rounds is flat at an aspect of 2π/5 = 1.2566, and plain knit in its dry-relaxed state is 1.2500. Eight every two rounds — a common instruction — is flat only at π/2 = 1.571, which no plain knit reaches, and so it is not a disc at all.

The washing crossover #

Plain knit does not have one aspect. A knit relaxes for as long as it is allowed, and its loop constants are published for three states: dry-relaxed, straight off the needles and rested; wet-relaxed, after a soak; and fully relaxed, after a soak and a tumble. The aspect in those three states is 1.2500, 1.2927 and 1.2791.

Both headline constructions have their flat aspect inside that band. So both of them change sides when the disc is washed — and they change in opposite directions.

Four wedges, knitted and rested, are dry-relaxed: each wedge is 91.7 degrees, the four close 366.7, and the extra 6.7 degrees ruffles the edge. Soaked, the aspect rises to 1.2927, each wedge shrinks to 88.6 degrees, the four close 354.6, and the disc rises into a cone 17 per cent of its radius high. Tumbled, it settles at 358.3 and 9.5 per cent.

Ten increases every two rounds does exactly the reverse. Dry-relaxed it grows 0.995 of flat and rises 10 per cent of its radius; soaked it grows 1.029 of flat and ruffles; tumbled, 1.018, still ruffled.

The same yarn, knitted into the same size of circle, lies domed one way and waved the other, and a wash swaps them. That is not a small claim about knitting and it follows from nothing but the three published constants and the geometry of a disc.

The ruffle side is not symmetrical with the cone side #

A disc that grows too little has somewhere to go: it closes up into a cone, a well-defined surface with a height that can be computed. A disc that grows too much does not. There is no cone with more than 360 degrees round its apex, and the extra circumference has to leave the plane as a wave — how many waves, and how deep, is decided by the fabric’s bending stiffness against its extensibility, which is a mechanics problem not solved here for a knit.

So the two failures are not mirror images, and the figures here treat them differently. A cone is drawn and its height is given; a ruffle is reported as an excess and nothing more. A ruffle is a claim about how much length has nowhere flat to go; where it goes is not in the model.

That asymmetry matters for practice. A small cone can be pressed flat by blocking, which stretches the fabric along its circumference — something a knit is good at, because nothing elastic opposes a knit’s extension. A small ruffle can only be pressed flat by compressing the circumference, which a knit resists by buckling. So a disc that errs is better off erring towards a cone, and the arithmetic says which construction does, in which state.

Eight every two rounds is a crown #

The most common instruction for a knitted circle in the round is eight increases every other round, and it does not make a circle.

Eight every two rounds grows 0.814 of flat in the fully relaxed state. That is a cone with a half-angle of 54.5 degrees, which rises 58 per cent of its radius. Read the other way, it is exactly the decrease rate used to close a hat’s crown — eight decreases every other round — and a hat’s crown is a dome. The instruction for a flat circle and the instruction for a domed crown are one instruction, run in opposite directions, and only the second is doing what it says.

Nine every two rounds is closer, at 0.916 and a rise of 40 per cent. Ten every two rounds is the first that reaches flat anywhere in plain knit’s band.

Six every round is a crochet rule #

The other rule that circulates is six increases every round. It comes from crochet, where the stitch’s aspect is different and it is correct.

In plain knit it grows 1.22 of flat in the fully relaxed state, 22 per cent too much circumference — a ruffle deep enough to be decorative. That is what makes the rule instructive rather than merely wrong. A rule for a flat circle is a statement about a stitch’s aspect, since $\Theta = m k_r / k$ sets the aspect it expects to $2\pi k/m$. Six every round expects an aspect of 1.047, which is nearly square; crochet stitches are near square, and knitted loops are not.

The same reading applies to every rule in a pattern book. Each one carries an implied aspect, and whether it is right for a particular fabric is one comparison with that fabric’s aspect.

Nothing about the yarn is in it #

Every quantity above is a count of loops times a spacing, and every spacing is the loop length over a constant. So the loop length cancels out of Θ exactly as it cancelled out of a tube’s circumference ratio and out of the heel’s turn: a four-wedge disc in a fine yarn at a tight loop and one in a coarse yarn at a loose loop are the same shape at different sizes.

That is a stronger statement than it looks. It says the choice of yarn cannot rescue a construction that is wrong for the fabric — knitting eight every two rounds in a thicker yarn makes a bigger crown, not a flatter circle — and it says the only thing that moves a disc’s flatness, once the construction is chosen, is whatever moves the loop’s shape. Two knits with one tightness factor are one knit found that the loop’s shape in plain jersey is one curve for every yarn; the relaxation state is the one thing left that moves it, and it moves it by three and a half per cent.

Three and a half per cent is a small number to hang a shape on. It is enough here because both headline constructions were already within that distance of flat, and because a cone’s height grows as the square root of its deficit, so the first per cent of misfit is the most visible one.

What a knitter can check with a tape measure #

The claim that a disc’s flatness is set by one ratio can be checked without any of this collection’s constants.

Measure a swatch: wales per ten centimetres and courses per ten centimetres. Their ratio, courses over wales, is the aspect — the same quantity Munden’s constants give, measured directly. Divide 2π by it and the result is the increases per round a flat disc worked in rounds needs; multiply 2π by it over two and the result is the number of wedges a sideways disc needs.

For a typical plain-knit swatch of 22 wales and 28 courses to ten centimetres the aspect is 1.27. That wants 4.9 increases a round, which is ten every two rounds, and four wedges. And it wants them for that swatch in that state only — the swatch measured again after a wash gives another aspect and another answer, which is the whole of the washing crossover in a form a knitter can see.

The model named #

The disc is a surface whose circumference at each radius is a count of loops. Worked sideways, the radius is a count of wales out from the centre and the circumference is a count of courses; worked in rounds, the radius is a count of courses and the circumference a count of wales. The loop constants are Munden’s, 5.0 and 4.0 dry-relaxed, 5.3 and 4.1 wet-relaxed, 5.5 and 4.3 fully relaxed, from this collection’s loop geometry; the aspect is their quotient. A cone’s height is the cosine of its half-angle over its slant radius, and the half-angle is the arcsine of Θ/2π.

The table is required, not shown. The flat aspect of four wedges and of ten increases every two rounds must each fall inside plain knit’s relaxed band, and the two must cross flat in opposite directions between the dry and the fully relaxed state; if either failed, the figure headed by that claim could not be drawn.

What was counted #

Seven constructions: three, four and five wedges of short rows each stopping one stitch further from the centre, and eight every two rounds, nine every two rounds, ten every two rounds and six every round. Each in the three relaxed states, which is twenty-one values of Θ/2π and twenty-one verdicts — cone, ruffle or flat — plus a cone height for each cone. The curves are the same formulas swept over the aspect from 1.15 to 1.40.

The seven are the constructions that turn up in pattern books for a flat circle or a crown, chosen before any was computed. Nothing was fitted, and the only published inputs are the loop constants.

What the picture cannot show #

A knit is not rigid, and every conclusion here is about the shape a disc would take if it had to keep its own dimensions. It does not: a knitted disc can be blocked, stretched, and worn on a head that forces a shape on it. The model gives the shape the disc is asked to take by its own loop counts — its natural curvature — and the size of the force needed to argue it into another shape depends on the fabric’s stiffness, which is not here.

A ruffle’s shape is not computed. The excess is exact; the waves it forms are not, for the reason given above.

The constants are relaxed plain jersey’s. A disc worked in garter stitch, rib or a lace pattern has loops of a different aspect, and composite structures do not inherit it from their parts. Every rule here would need its own measured aspect, and the swatch method gives it.

And the centre is idealised. The first few wedges or rounds near the centre are not continuous surfaces at all — a wedge at a radius of two stitches is a pair of loops — so the model is about the disc beyond the first handful of stitches out.

The generalisation is Gauss’s #

A surface’s curvature can be read off from inside it, without looking at it from outside, by measuring how circles grow. That is Gauss’s result, from 1827, and it is exactly what knitters do by instinct when they count stitches round a circle.

A knitted fabric is a surface whose metric — the distances within it — is set by counts of loops and a constant. That makes every knitted shape a statement about curvature in Gauss’s sense: the heel concentrates it along one line, the disc spreads it over a surface, and the constant converting counts into distances is the loop’s aspect in both. A woven dart gets its curvature by cutting cloth away and seaming the edges; a knit gets it by counting, and cannot get it any other way.

Who found it, and when #

The disc’s condition is elementary differential geometry. The use of knitting and crochet to make surfaces of chosen curvature is Daina Taimiņa’s, whose crocheted hyperbolic planes of 1997 are the famous example: a crochet surface that grows by a fixed ratio each round is a model of the hyperbolic plane, because its circumference grows exponentially rather than in proportion.

The rules for flat circles in knitting are craft knowledge and vary between sources. That knitted loops have a published aspect and that it moves with relaxation is Munden’s, from 1959, and this collection’s loop geometry carries it.

What is new here is putting the aspect into the disc, which turns a set of rules that disagree into one condition each rule can be checked against — and finds that two of them sit exactly on the edge of the relaxed band, on opposite sides of it.

Still open: the knitted sphere #

A disc is a surface of zero curvature built from a count that almost always misses. A sphere is a surface of constant positive curvature, and it is what a knitted ball, a toe and a hat’s full crown approximate — usually as a stack of wedges tapering at both ends, like the gores of a globe.

The arithmetic is the same and the answer should be sharper still. A sphere knitted in m gores needs each gore’s width to follow the sine of its latitude, which a whole-stitch count can only approximate in steps, and the number of gores it needs to close is 2π over the gore’s angle at the equator — another count set by the loop’s aspect. Whether a knitted sphere can close at all without a flattened pole, and how many gores plain knit wants, is the next calculation, and it is the one where the heel’s single line and the disc’s single number have to be combined.