A fashioned edge has a quantised angle

Everything this site has said about making a flat cloth fit a shape has been about cutting. A bias-cut square costs exactly twice its own area. A hemisphere costs one full turn of removed angle. A dart is where the shear ran out. All of it takes the fabric as given and removes material.

A fully-fashioned knitted panel is made the other way round. Nothing is removed. The machine transfers a loop from one needle onto its neighbour, and from that course on it knits one fewer wale — so the panel narrows by a whole wale at a whole course, and the edge is a staircase rather than a line.

That has a consequence with no analogue in cutting. A cut edge can be at any angle whatever; a fashioned edge can be at j wales in k courses and at nothing between two fractions. The set of achievable angles is discrete, and it is much smaller than a knitter would guess.

The angle, and the constant in it #

A relaxed plain knit has k_c / ℓ courses per centimetre and k_w / ℓ wales per centimetre, where ℓ is the loop length and the two constants are Munden’s. So the wale spacing is ℓ / k_w and the course spacing is ℓ / k_c, and the loop length cancels out of their ratio.

An edge stepping j wales in k courses therefore makes an angle from the wale direction of

θ = arctan( j·k_c / (k·k_w) ) = arctan( j·k_r / k )

where k_r is the ratio of the two constants, published separately as a fourth constant, and equal to 1.279 in the fully-relaxed state.

Everything that a knitter chooses has left the expression. Not the yarn count, not the fibre, not the machine gauge, not the loop length, not the fabric’s density. Two garments knitted in different yarns on different machines have their fashioned edges at exactly the same angles, and the angles in between are unavailable to both.

That is a strong claim of the kind worth checking rather than asserting, so it is: the angles are recomputed from first principles on four plain knits whose loop lengths span a factor of four and whose stitch densities span a factor of sixteen, and the four lists are required to agree to nine decimal places.

Eighteen angles, unevenly spaced #

A fashioning machine transfers at most one or two wales at a side in a course — more than that distorts the fabric badly — and a garment panel is not so long that fractions with denominators past a dozen are useful. That gives eighteen distinct angles.

They run from 6.08° at one wale in twelve to 68.65° at two wales in one, and they are conspicuously not evenly spaced. They crowd towards the wale — 6.08, 6.63, 7.29, 8.09, 9.08 — and thin towards the course: 32.60, 40.45, 51.98, 68.65. The widest gap is 16.67 degrees, between one wale in one course and two wales in one.

And above 68.65° there is nothing at all. A fashioned edge cannot approach the horizontal, because the shallowest possible staircase is the steepest transfer rate the machine allows.

What a garment asks for #

Set the available angles against the edges a garment pattern actually contains and the gaps land where the garment is.

A fitted waist wants about 6° from the wale and gets 6.08 — an error of 0.08 degrees. A sleeve underarm wants 18 and gets 17.73. Those are exact for any purpose.

A raglan front wants about 35° and the nearest available is 32.60, which is 2.4 degrees out. A set-in sleeve head wants 55 and gets 51.98, three degrees out. And a horizontal neck wants 90 and the nearest fashioned edge is 68.65, 21.35 degrees out of reach, which is not an approximation of anything.

The gaps widen as the edges steepen, and a shoulder is where they stop being a rounding error: the difference between what a pattern asks for and what the machine can cut is several degrees rather than a fraction of one.

There is nothing at all beyond the upper bracket of that pair, and the reason is mechanical rather than arithmetical: the machine runs out of transfers it can make in one course, not out of fractions it could name.

What was counted, and how #

The angle set is enumerated over fractions rather than derived from a formula. Every j up to the machine’s transfer limit and every k up to a panel’s useful length is reduced to lowest terms, duplicates are discarded by their reduced form, and the angles are computed from the loop aspect and sorted. The widest gap is read off the sorted list.

The loop aspect itself is checked three ways before it is used. It is computed as the quotient of the courses and the wales a relaxed fabric has, at a stated loop length; it is required to equal the quotient of the two published constants exactly; and it is required not to move when the loop length is doubled. The published fourth constant is quoted beside it rather than substituted for it, because Munden’s four constants are separately fitted and do not satisfy their own identities exactly — 1.279 against a published 1.30 in this state, which is the size of disagreement a set of regression constants has.

The universality claim is checked by recomputation rather than by algebra: four fabrics, each with its own wale spacing and course spacing in centimetres, each giving the full list of angles, all four required to agree to nine decimal places. And the fabrics are required to be genuinely different, by asserting that their stitch densities span a factor of more than four — because a universality claim tested on four nearly identical fabrics is not a test.

Two assertions guard the shape of the result. No fashioned edge may be horizontal, which is what makes the neck unreachable. And the widest gap must not be at the shallow end of the list, which is the claim that the crowding is where it is said to be.

The gaps are not spread evenly either #

The list crowds at one end, and it is worth saying how severely, because the practical consequence is not that fashioning is coarse but that it is coarse in exactly one place.

Take the eighteen angles in order and read off the distances between them. At the shallow end they are fractions of a degree: 6.08, 6.63, 7.29, 8.09, 9.09, so half a degree, two thirds, four fifths, one. By the middle they are two and three degrees. The last four gaps are 5.49, 7.85, 11.53 and 16.67, and those four gaps between them account for more than half of the whole 62.6-degree range that fashioning can reach, with three angles inside them.

The reason is the arctangent and the reciprocal together. The available fractions are j/k with j at most two, so at the shallow end consecutive fractions differ by about 1/k² and the angle differs by about kᵣ/k² radians — half a degree at k = 12, and shrinking as the square. At the steep end there are only four fractions left above one half, and they are 1/2, 2/3, 1/1 and 2/1, which are not close to anything.

So the resolution of a fashioned edge is a function of the angle itself, and it improves as the fourth power of nothing anybody chooses: it improves because a longer panel admits a larger denominator, and a larger denominator is only available where the numerator is small. A garment that wants a shallow edge can have it to a hundredth of what it asked for. A garment that wants a steep one cannot have it at all.

What the mixed rate actually costs #

Alternating two rates buys a mean direction anywhere between them, and it is worth being precise about what is given up, because the answer is not accuracy.

A staircase of a single rate j/k departs from the straight line it approximates by at most one wale, and it returns to the line every k courses. A mixture of two rates in some proportion has a mean direction between them and returns to its line only at the end of the whole repeating pattern of decreases, which is the sum of the two denominators or a multiple of it. The departure is still bounded by a wale — nothing can be further from the line than one step — but the wavelength of the wander has grown by the same factor the angular resolution improved by.

That is the trade, and it is the trade a lattice always offers. A finer mean angle is bought with a longer wave in the edge, at fixed amplitude. On a fine-gauge panel one wale is a fraction of a millimetre and nobody sees the wave; on a heavy hand-frame garment at three wales to the centimetre it is three millimetres of visible scallop, and a mixed rate is avoided for that reason rather than for any arithmetic one.

Nothing about that is special to knitting, and it is the same bargain a halftone screen strikes, or a stepped gear train, or any other device that reaches an unavailable ratio by alternating two available ones: the mean is exact and the instantaneous value is never the mean.

The amplitude is what a seam feels and the wavelength is what an eye sees. A seam sewn along a mixed-rate edge is joining two edges whose waves need not be in phase, so the seam takes up the difference — which is why fashioned panels are made in mirrored pairs from the same instructions, and why a pattern that specifies decreases as a sequence rather than as a rate is specifying the phase as well as the angle.

What a knitter does about the gaps #

The gaps are real and garments get made anyway, so it is worth setting out what the trade does with them — because each remedy is a different way of leaving the lattice.

Alternating two rates. An edge wanted between one wale in two courses and one in three can be worked as two of the first and one of the second, repeatedly, which gives a mean angle between them. That is not a new angle in the sense of this essay: the edge is a staircase with two different tread lengths, and locally it is still one of the eighteen. What it buys is a mean direction anywhere in the range, at the price of an edge that is no longer a straight line but a very slightly wavy one. It is what a pattern book means when it says “decrease one stitch every third row four times, then every second row six times”.

Holding. A shoulder is shaped by leaving stitches on their needles and knitting shorter and shorter courses across the rest, then knitting them all off at the end. That produces an edge with no transfers in it at all, at an angle set by how many stitches are held per course — which is a different quantisation with a different lattice, and one whose shallow end is where the transfer lattice’s is sparse. The two mechanisms are complementary, and the fact that garment shaping uses both is the practical form of everything above.

Cutting. Cut-and-sew knitwear abandons the lattice entirely and takes any angle at all, paying for it in waste and in an edge that will run unless it is secured. Which is the trade the whole field is about, and it is the same trade the woven side makes between a stepped figure boundary and a cut one.

Where the model stops #

Munden’s constants are for a relaxed plain knit and nothing else. A rib, an interlock, a purl or anything with a tuck in it has its own dimensions, and this site has already found that the constants do not compose — the yarn in a composite structure adds exactly and the size it relaxes to does not. So the eighteen angles are plain jersey’s. Another structure has another set, computed the same way from constants nobody has measured.

The relaxation state moves the aspect a little. Dry-relaxed, wet-relaxed and fully-relaxed give aspects within three and a half per cent of each other, so the angles move by a fraction of a degree between states. That is far smaller than the gaps and does not change any conclusion, and it is stated because a claim of exactness that quietly held only in one state would be worth nothing.

The transfer limit is a machine property and is an argument. Two wales at a side in a course is ordinary; some machines do more, and a machine that could transfer four would reach 79 degrees and would still not reach 90.

And nothing here says what the edge looks like. A transferred loop shares a needle with the one already on it, and what results is the fashioning mark — a visible line of doubled stitches that is sold as a mark of quality. Whether it reads as a hole, a thickening or a decoration depends on the yarn and the tension, and none of that is here. The marks in the drawings are positions.

The generalisation #

This is the same shape as a woven outline’s staircase and the comparison is worth making because the two mechanisms have nothing else in common.

In both cases a shape is approximated on a lattice, the achievable directions are the reduced fractions the lattice admits, and those directions are a Farey set — dense at the ends and sparse in the middle. In both cases the lattice’s cell has a physical size, so the quantisation is in millimetres rather than in units of anything.

What is different, and what makes the knitted case sharper, is that the lattice’s aspect ratio is a universal constant. A woven cloth’s cell is its repeat divided by its setts, and a designer changes it by changing the sett. A knitted fabric’s cell is ℓ/k_w by ℓ/k_c, and the ratio is k_r whatever anybody does. So the woven case has a resolution that depends on the cloth and the knitted case has a shape that does not.

The general lesson is that a quantisation inherited from a manufacturing operation can be far coarser than the tolerances anybody is working to, and can be invisible because the operation is described in its own units — wales and courses — rather than in the units the specification is written in, which are degrees.

Who found it, and when #

Fully-fashioned knitting is Victorian and the shaping rates are in every knitwear technician’s tables, given as “one in two”, “two in three”, “one in four” — which is exactly the fraction j/k above, quoted as a rate rather than as an angle. The practice of shaping a shoulder by holding rather than by transferring is universal and is taught as a rule.

What is not usually written is the reason the rule exists, which is that the shoulder is past the last fraction. And what appears not to be written anywhere is the universality: the tables are given per machine and per gauge, as though the achievable set depended on them, and it does not. Two of Munden’s constants divide out and the third is the answer.

Munden’s work on the dimensional properties of plain knitted fabrics is from the 1950s and 60s, and the constants are his. The observation that their ratio is the loop’s aspect, and therefore the thing that quantises a fashioned edge, is this site’s — and it is available only because the geometry was already here for a different purpose.

Where the ladder goes next #

The next rung asks what happens when the needle count cannot be changed at all, which is the situation on a circular machine knitting a seamless tube. The only free quantity left is the loop, the loop is bounded at both ends by what the yarn will make, and the taper that leaves is 18.8 per cent and no more — paid for in a fabric half again as dense at the narrow end.

Sideways, the same shape reached by cutting is the bias cut and the dart, which have no quantisation at all and waste material instead; the geometry the angles come from is the loop’s own dimensions; and the same Farey structure in a woven figure is the staircase a woven outline is.