A garment is cut dry and worn wet

A cutter marks a pattern on cloth, adds a shrinkage allowance, and cuts. The allowance is a percentage, it comes off a specification, and it is one number.

The thing it is allowing for is at least two numbers in two directions, and where those two differ it is also a rotation that no percentage can express and no tape measure can find.

The two directions are not the same number #

For the balanced cloths in this collection’s table they very nearly are. The batiste shrinks 2.54 per cent across and 2.56 along; the muslin 3.09 and 3.14; the voile 0.93 and 0.94. Within the resolution of the locus scan, those are isotropic.

That is not a coincidence and it is not a general fact. It is what a balanced construction does: equal counts in warp and weft at nearly equal setts, so the two systems are nearly interchangeable and whatever happens to one happens to the other.

The poplin is the unbalanced one — 15 tex warp at 32 ends per centimetre against 20 tex weft at 22 picks — and it shrinks 3.67 per cent across and 1.82 along, a ratio of 2.01.

So the rule is easy to state: a shrinkage allowance is one number for a balanced cloth and two for an unbalanced one, and the trade’s habit of quoting one is a habit formed on balanced cloths.

The bias, where it stops being arithmetic #

A tape measure laid along the warp reads one shrinkage and along the weft the other. Laid at forty-five degrees it reads something in between, and the natural assumption is that it reads the average.

It very nearly does. For the poplin the mean of 3.67 and 1.82 is 2.745, and the true bias figure is 2.74. Over the range of shrinkages a cloth actually undergoes the two agree to a hundredth of a point, because the exact expression differs from the mean only at second order.

That is the boring half of the answer. The interesting half is that a line cut at forty-five degrees does not stay at forty-five degrees.

Its component along the warp shrinks by one amount and its component along the weft by the other, so its angle changes:

$$\theta' = \arctan\!\left(\frac{(1 - s_{\text{length}})\sin\theta}{(1 - s_{\text{width}})\cos\theta}\right)$$

For the poplin that is 0.544 degrees of rotation. Across a panel a metre long, that is 9.5 millimetres of skew.

Half a degree is a great deal #

Nine and a half millimetres across a metre does not sound like much until it is put where it lands.

A bias-cut skirt panel is cut so that its centre line runs at forty-five degrees to both grains, precisely so that it drapes evenly and hangs in a circle. If the two grains shrink differently, that centre line rotates and the panel is no longer symmetrical about it: one side hangs longer than the other and the hem goes off-level.

A seam joining two panels cut at opposite bias angles rotates in opposite directions, so the seam twists. That is the classic complaint about bias-cut garments — that they hang beautifully until they are cleaned — and it is usually attributed to the fabric relaxing unevenly, which is right, and left there.

And a tape measure cannot find it. Measure the panel along the warp: correct. Along the weft: correct. Diagonally: correct to a hundredth of a point. The panel has changed shape and every length measurement on it is what the allowance predicted, because the change is a shear and shears preserve lengths along the directions they act between.

A shear is exactly the deformation a length measurement is blind to, which is a fact the bias established for a cloth being pulled and applies here unchanged to a cloth being washed.

Which cloths do it and which do not #

The rotation is asserted rather than described, and in two halves because either alone would say nothing.

A cloth whose two shrinkages agree to a thousandth must rotate by under a twentieth of a degree — the balanced four in this collection’s table all come in under 0.013 degrees, which is a millimetre across two metres and is nothing. A cloth whose shrinkages differ by more than half a point must rotate measurably, and the poplin does.

So the practical test is a simple one and it is available at the specification stage: if the two grain shrinkages differ, the bias will skew, by roughly the difference in radians. For the poplin the difference is 1.85 percentage points, which is 0.0185, which is 1.06 degrees — and the true answer is 0.544, half of it, because the rotation at forty-five degrees is half the difference to first order.

That factor of a half is worth having as a rule: the bias rotation in degrees is about a quarter of the difference in the two shrinkages in per cent.

The rotation at every angle, not just at forty-five #

The half-of-the-difference rule is quoted above for a line cut at forty-five degrees. The general expression is worth having, because it decides how far off the bias a cutter has to go to escape the skew, and the answer is discouraging.

Uneven shrinkage is a strain of one amount along the warp and another along the weft. Split it into a uniform part, which changes no angles, and what is left: a pure shear whose engineering magnitude is the difference of the two shrinkages. A line at angle θ to the warp then rotates by

half the shrinkage difference, times the sine of twice the angle.

That reproduces the worked example — 1.85 points of difference is 0.0185, halved is 0.0093 radians, and at forty-five degrees the sine is one, giving 0.53 degrees against the scan’s 0.544.

And it says what the angle is worth. The rotation is greatest on the bias and vanishes on both grains, which is expected. What is not expected is how flat the maximum is. At thirty degrees the sine of sixty is 0.87, so a panel cut well off the bias still takes seven eighths of the full skew. Halving it means reaching sin 2θ = 0.5, which is fifteen degrees from a grain, or seventy-five.

So there is no safely-off-grain. Anything cut between fifteen and seventy-five degrees takes at least half the maximum skew, and that band is where every panel cut for drape lives.

One further number falls out for the case the essay names. Two panels cut at plus and minus forty-five rotate in opposite senses, so the angle at the seam between them changes by the whole difference rather than half of it — 1.06 degrees for the poplin. The seam twists by twice what either panel does, which is why the seam is where the fault is seen.

The allowance a cutter actually needs #

Putting this together, an honest allowance has four parts and the trade quotes one.

A warp-way figure and a weft-way figure, because they differ in an unbalanced cloth and the difference is a factor of two in the worked example here.

A statement of which shrinkage is being quoted. Two shrinkages share one tape measure: hygral shrinkage is recoverable, immediate and geometric, and relaxation shrinkage is permanent and needs agitation. A pre-shrunk cloth has had the second taken out and still has all of the first, so its label figure and its behaviour in a wash are about different quantities.

A skew figure for anything cut off-grain, which is the difference of the first two and is currently quoted nowhere.

And a note about the state. A garment measured wet is a different size from the same garment dry, permanently, because the hygral part comes back on drying and cannot be finished out. A shirt that fits when dry is a shirt that is two per cent tighter for the hour after a wash, for the rest of its life.

None of that is exotic and none of it needs a new measurement — the two grain figures are what a standard shrinkage test already produces, and it reports them separately before somebody averages them.

Why an unbalanced cloth is the common case #

It would be comfortable to conclude that this only matters for the odd cloth, since four of this collection’s five wet-capable cloths are near enough isotropic.

The table is not representative in that respect and it is worth saying so. This collection’s eight cloths were chosen to span a range of cover and count, and seven of the eight are balanced or nearly so — equal counts in warp and weft. Real shirtings, suitings and sheetings very often are not: a poplin’s whole character is a fine dense warp and a coarser open weft, and that is a common construction rather than an exotic one.

More generally, almost every woven cloth is set more densely in the warp than in the weft, because a loom’s warp is under tension and its weft is beaten in, and this collection’s own table shows it — all eight have more ends than picks. That asymmetry alone is enough to make the two shrinkages differ; it is the counts being equal that makes them nearly agree, and the counts are equal in seven of eight here and in rather fewer cloths in the world.

So the poplin is the representative case and the other four are the special one, which is the opposite of the impression the table gives.

What the fibre choice buys #

The whole effect scales with the swelling, so the fibre is the largest lever a specifier has.

Polyester swells by about a thousandth of a per cent, so a polyester panel has no hygral shrinkage, no skew and no wet-tight hour. A polyester-cotton blend has roughly the cotton fraction’s, so a 65/35 shirting has about a third of an all-cotton one’s.

Viscose swells thirty-five per cent, which is 1.75 times cotton’s, and is correspondingly worse in every respect — which is the practical content of a viscose garment’s dry-clean label.

And linen only shrinks. Flax’s axial swelling is a thousandth against cotton’s twelve thousandths, so the term that opens an open cloth out is absent, and a linen cloth of any construction contracts. That is a fact about a fibre showing up as a fact about a garment, and it matches linen’s reputation exactly.

The order of operations, which decides the size #

There is one more thing a cutter controls and it changes the answer more than any of the above: when the cloth is wetted relative to when it is cut.

Cut dry and sew, and the garment carries the whole allowance and the whole skew. Wet the cloth, dry it and then cut — which is what a pre-shrinking works does and what a careful home sewer does by washing the fabric first — and the relaxation part is gone before the pattern is marked.

The hygral part is not, and cannot be, because it is not a departure from equilibrium: there is nothing to take out. So pre-washing removes one of the two shrinkages and the whole of the relaxation skew, and leaves the hygral shrinkage and the hygral skew exactly as they were.

That is a sharp practical statement and it is testable. A pre-washed cloth cut on the bias should still skew in a wash, by the hygral amount and no more; a cloth cut straight from the bolt should skew by the sum. If pre-washing removed all of it, the argument in this rung would be wrong.

It also explains why the trade’s remedy — pre-shrinking — helps a great deal with size and not much with bias distortion. The two parts have different sizes in the two effects: the relaxation part dominates the length change and the two parts are comparable in the anisotropy, because relaxation is anisotropic for a different reason and not necessarily by the same ratio.

What was counted, and how #

The bias rotation is trigonometry on the two shrinkages and nothing else, so what is asserted is the relation rather than the size.

A cloth whose two grain shrinkages differ must rotate, and one whose shrinkages agree must not. Both halves, at a floor of a tenth of a degree for the first and a twentieth for the second. A model that produced a rotation for a balanced cloth would have a sign error or a stray asymmetry in it, and this is what would say so.

And the two shrinkages themselves carry the assertions of the rung below: the closure margin agrees with its closed form, the wet minimum is geometric for a balanced cloth, and suppressing the axial swelling makes every cloth shrink.

The skew in millimetres is the rotation times the panel length and is not separately asserted, because there is nothing in it a check could catch.

Where the model stops #

Only the hygral half is computed. The relaxation half is anisotropic too — the warp shrinks more — and the two anisotropies do not have to point the same way. Combining them is straightforward arithmetic and needs the loom tensions, which are chosen rather than measured, so it has not been done here.

No plasticity, so this says what a garment does when wet and not what it is left with.

The panel is treated as free. A real panel is sewn to other panels and constrained by seams, so it cannot shear freely; instead the constraint puts the shear into the seam as pucker, which is exactly what the wool trade’s hygral-expansion literature is about. This collection has an arithmetic for what holds a thread in a seam and has not joined the two.

And the rotation is computed for a flat panel. A garment is not flat, and a bias panel on a body is already sheared by being made to fit a curved surface. Adding a wash-induced shear to that is a real problem and it needs the drape machinery rather than this arithmetic.

What the skew is worth against the seam allowance #

It is fair to ask whether nine and a half millimetres across a metre is worth the fuss, since a seam allowance is ten to fifteen millimetres and a hem is more.

Two reasons it is.

The skew is a rotation and an allowance is a length. Adding five millimetres all round does not fix a panel that has rotated half a degree: the panel is the right size and the wrong shape, so the extra cloth is in the wrong place. A hem that has gone off-level by nine millimetres cannot be corrected by having cut the panel nine millimetres longer.

And it accumulates across a garment. A skirt made of six bias panels has six rotations, alternating in direction where the panels are cut from alternating layings, so adjacent seams disagree by twice the figure. Nineteen millimetres between two panels at a hem is visible from across a room, and it is the exact complaint that bias-cut garments attract.

Against that, the allowance’s real job — making sure the finished garment is not too small — is served perfectly well by one number, and the two grain figures differ by less than a typical allowance in every cloth here. So the scalar allowance is right for the job it was invented for and wrong for the one it is now being asked to do, which is the ordinary fate of a good specification.

The generalisation #

A specification that quotes a scalar for a tensor quantity has already lost something, and the lost part is usually the one nobody measures.

A shrinkage is not a number; it is a deformation, and a deformation of a sheet has three independent components — two stretches and a shear. Quoting one number keeps a trace of the two stretches and discards the shear entirely.

The shear does not appear because the standard test measures lengths, and lengths are exactly what a shear preserves. So the instrument and the specification agree with each other perfectly and are both blind in the same direction, which is the most durable kind of blindness there is.

The remedy is to ask what deformation the quantity is a summary of, and then to ask which components of it the measurement can see. Here that question has a one-line answer and it points straight at the missing figure.

Who found it, and when #

That bias-cut garments distort in laundering is old trade knowledge and is why they are usually dry-cleaned. That the cause is differential shrinkage in the two grain directions is the standard explanation and is right.

Hygral expansion in wool and the seam pucker it causes are a well-studied problem in tailoring, with standard tests and standard remedies, and the mechanism there is exactly this one.

Whether the rotation has been written down as a rotation — with the quarter-of-the-difference rule and a millimetre figure across a panel — this collection does not know. It is elementary once the two shrinkages are separate numbers, and the reason it may be unusual is that they are usually not.

Where the ladder goes next #

Into the seams, where a panel that wants to shear and is sewn to one that does not is a pucker rather than a skew — which is a question about a seam’s own stiffness and is the piece this collection has not joined.