What a thickness gauge reads on a knit

This collection has a rule about thickness measurements and it was written for woven cloth: a thickness gauge reads a maximum rather than a mean, because its foot lands on the highest crowns and never reaches the valleys between them. That is why Peirce’s circular section over-predicts every measured woven thickness and why the over-prediction is not an error in the geometry.

A knitted fabric has now been given a structural thickness too — two yarn diameters, from the interlacing, with nothing fitted. The same question arises immediately and the answer is not the same, because the surface a gauge is landing on is a different kind of surface.

The structural number #

0.334 millimetres for a 20 tex cotton, and the same for the same yarn at any gauge, because the interlacing passes at a diameter whatever the loop length.

That number is a lower bound on any measurement, for three reasons that stack rather than compete.

It is worth being precise about what “the same at any gauge” means, because it is the surprising half. Change the loop length and the fabric changes in every plan dimension — courses, wales, stitch density, areal weight all move — and the thickness does not move at all, because the thickness is set at the one place in the loop where two lengths of yarn pass over one another and that crossing is two diameters deep whatever else the loop is doing. A jersey at a 2.5 mm loop and the same yarn at 4 mm are different fabrics by every measure a mill records and are the same fabric through their own thickness.

And it is a lower bound rather than an estimate. Nothing in a real fabric can be thinner than its own interlacing, so any reading below 0.334 mm is a reading of a fabric that has been compressed — which is to say, a reading taken with a gauge. That is what makes the number useful: it is not a prediction to be compared against a measurement, it is a floor the measurement has to sit above, and a measurement that does not is evidence about the instrument.

One: the surface is entirely crowns #

A woven cloth has flat stretches. Between two crossings a thread runs comparatively straight and comparatively level, and a gauge foot large enough to span several crossings rests on the crowns while bridging over those stretches.

A knitted fabric has no flat stretches at all. Every part of its surface is either a crown — the top of a head or a sinker arc — or a leg sloping away from one. The tilt of the loop’s plane means that even the “flat” run between two interlacings is climbing.

So the crown statistic that makes a woven measurement a maximum is not a correction on a knitted fabric; it is the whole surface. A gauge lands on the highest crowns, and on a knit the highest crowns are the fabric.

That pushes the reading up relative to a mean, but not relative to the two diameters — the two diameters already run crown to crown. This first effect is smaller on a knit than on a woven cloth, which is the opposite of what the phrasing suggests.

There is a second-order consequence of that which is worth having. Because the surface is all crowns, a gauge foot of any size lands on very nearly the same statistic whatever its area — where on a woven cloth a small foot and a large one read differently, because a small one can drop into a valley. So a knitted thickness ought to be less sensitive to the foot’s size than a woven one and more sensitive to its load. That is a testable asymmetry and nobody here has tested it.

Two: the canopy, which is the large one #

Every spun yarn has fibre standing off it. This collection computes that layer as a population rather than a thickness: a density falling away from the yarn’s surface with a characteristic depth, from which a canopy depth follows.

On the cotton yarns here that depth is a few tenths of a millimetre — comparable with the entire structural thickness of a jersey. A raised or brushed fabric’s is several times more.

A thickness gauge under a light load does not push through a canopy. It rests on it, or on some fraction of it, and the fraction depends on the load. So the measured thickness of a knitted fabric under a light load is the structure plus most of two canopies, one on each face.

That is the largest of the three effects by a wide margin, and it is the reason a measured knitted thickness is not close to two yarn diameters.

Three: the load has already done work #

A gauge measures at a stated pressure, typically around a kilopascal, and a kilopascal is not nothing.

The fabric’s own structural resistance is fifteen kilopascals of initial slope, so a kilopascal has taken a few per cent off the structure before the reading is taken. And the canopy’s resistance is far lower than the structure’s — a fibre standing off a yarn bends at a touch — so most of the canopy has already been flattened at the same load.

So the load is doing two opposite things to the reading. It is compressing the canopy, which lowers the measurement toward the structure, and it is compressing the structure, which lowers it below. Which dominates depends on the load, and that is why a knitted fabric’s measured thickness is far more load-dependent than a woven cloth’s.

A knitted thickness quoted without its load is not a measurement.

What the reading is therefore a measurement of #

Putting the three together: a light gauge on a smooth jersey is measuring the hair layer, with the structure underneath as a floor. A heavy gauge is measuring the structure, with the hair pushed aside and the loops beginning to move.

Neither reading is wrong and neither is the number this collection predicts. The prediction is of a structural thickness — where the yarn is — and the gauge measures where the fabric’s resistance first exceeds a threshold.

That is the same distinction the woven side draws, and on a knit it is much sharper because the canopy is a larger fraction of the whole.

The measurement that would separate them #

Run the gauge over a range of loads and plot the thickness against the pressure. Two features should appear.

At low load, a steep fall: the canopy collapsing, over a pressure range of a few hundred pascals, with a large change in thickness for very little load.

Then a knee, and a much flatter stretch: the structure resisting at its own fifteen kilopascals per unit strain.

The knee is the structural thickness. Extrapolating the flat part back to zero load gives the number to compare with two yarn diameters, and it is a far better comparison than any single reading.

This collection’s woven side already uses the same trick in reverse: it inverts a measured thickness against a compression model to recover the pressure the crossings are carrying. The knitted version has not been run.

Why the independence prediction is the sharper test #

The structural thickness does not move with the gauge: the same yarn knitted at 3.0, 3.5 and 4.5 millimetres of loop is predicted to be exactly as thick at all three.

A measured thickness almost certainly will move, and the direction is informative. If the looser fabric measures thicker, the extra is hair and crowns rather than structure, and the amount says how much of the reading is not the fabric. If all three measure the same, the canopy contribution happens to be constant across the gauges, which would be a coincidence worth knowing about.

Three swatches and a gauge, and the result is informative either way. That is a much better experiment than measuring one fabric and comparing it to one prediction, because a single comparison can always be explained away.

What the yarn’s own compression does #

There is a fourth effect and it works the other way from the first three, which is the reason it is worth separating rather than folding into them.

A flattened thread is a record of a force, and at a knitted fabric’s crossings the yarn is under the through-thickness contact force at all times — 7.8 millinewtons a stitch even at rest. So the yarn at the crossings is already slightly flattened before any gauge arrives, and a flattened yarn is wider than a circular one of the same area rather than deeper.

That lowers the structural thickness below two diameters. It is much the smallest of the four effects, because the contact force is small and a yarn’s transverse stiffness is not, and this collection has no model of a knitted crossing’s flattening to put a number on it. It is named so that the list of what a measurement contains is complete.

What a rib does to all of this #

A rib’s structural thickness is the bed gap plus a diameter, which for an ordinary setting is two to three times a jersey’s. The canopy is unchanged — it belongs to the yarn, not to the structure.

So the hair is a much smaller fraction of a rib’s reading than of a jersey’s, and a rib’s measured thickness should be much closer to its structural one. That is a second prediction from the same argument and it points the other way from the first, which makes the pair harder to satisfy by accident.

It also means the two fabrics’ measured thicknesses are not comparable in the way their structural ones are: the ratio of the readings is smaller than the ratio of the structures, and by an amount that depends on the yarn’s hairiness.

The general rule, restated for a knit #

The woven version of this rule says a gauge reads a maximum, not a mean. The knitted version has to be said differently:

A gauge reads a canopy, then a maximum, then a structure, in that order as the load rises.

Three regimes rather than two, because a knitted surface has a hair layer standing off a surface that is entirely crowns. On a woven cloth the first two regimes overlap enough to be treated as one; on a knit they do not.

Why this matters beyond bookkeeping #

Three quantities in this collection are computed from a thickness, and each of them would be wrong by a different amount if the wrong thickness went in.

Warmth, which is a thickness over an effective conductivity. Feeding a gauge’s reading in raises a jersey’s resistance by whatever fraction of the reading is hair — and that is not simply an error, because the hair really does hold still air. It is a different quantity: the fabric’s resistance including its canopy.

Compression, where the relaxed thickness is the point the curve starts from. Start it at the gauge’s reading and the first part of the curve is the canopy’s, not the structure’s.

And areal density per unit thickness, which is how a fabric’s bulk is specified in trade. That one is pure bookkeeping and is entirely at the mercy of which thickness was used.

What this does not settle #

How deep the canopy actually is on a knitted fabric. This collection computes hair populations from yarn parameters and has done so for woven cloths. Nothing here has been measured on a knit.

Whether the crown statistic differs between the two fabrics. It ought to — a knitted surface’s height distribution is a different distribution from a woven one’s — and it has not been computed.

And what a standard test is really specifying. The gauge loads in the standards were chosen for woven cloth, and there is no reason to think a load appropriate for a poplin is appropriate for a jersey.

The same problem on the woven side, for scale #

It is worth putting the knitted difficulty beside the woven one, because the woven one is measured and the difference is instructive.

This collection holds eight woven cloths with measured thicknesses, and for every one of them Peirce’s circular section over-predicts: the model says thicker than the gauge reads, by fifteen to seventy per cent. The direction is the opposite of the knitted case, and the reason is that a woven cloth’s threads are pressed hard together at their crossings and flatten, so the circular section over-states the crimp height.

A knitted fabric has almost no force at its crossings by comparison — 7.8 millinewtons through the thickness against a woven cloth’s hundreds — so its yarn barely flattens, and the structural prediction is not over-stated for that reason. What over-states the measurement instead is the hair.

Two fabrics, two discrepancies, opposite signs, different causes. That is a good sign for both accounts: a single systematic error in the diameter or the packing would push both the same way.

Why this is the sharpest exposure on the ladder #

Most of what this collection computes about a knitted fabric is a force, and a force here is the free end of a bracket a hundred and thirty wide. Nobody has measured a knitted contact force and nobody is likely to.

A thickness is not like that. It is a length, it is measured routinely, it is in every specification, and the prediction is bare: two yarn diameters, no fitted constant, no bracket.

So the thickness is where this account is most likely to be shown to be wrong, and the three effects above are the reason a naive comparison would show it wrong for the wrong reason. The load sweep is what turns an easy refutation into a real test, and that is the whole argument of this rung.

What is genuinely new here #

One distinction and one prediction.

Three regimes rather than two. A knitted fabric’s measured thickness passes through a canopy, a crown maximum and a structure as the load rises, and a single number quoted without a load is a reading in an unnamed regime.

And the knee is the structural thickness. A load sweep separates the hair from the fabric without needing either to be modelled, which turns an awkward comparison into a measurement.

What the pictures cannot show #

The section on this page draws the structure and nothing else: two centre lines, two radii, a mid-surface. There is no hair in it, no crown statistic and no gauge foot, and those are between them most of what a measurement sees.

A drawing that included the canopy would be a drawing in which the fabric was a thin dark band in the middle of a pale cloud, and that would be the honest picture of what a gauge lands on.

Which of the four effects is largest #

Putting the four in order matters, because a reader taking one measurement away should know what dominates it.

The canopy, by a wide margin. A hair layer a few tenths of a millimetre deep against a structure a third of a millimetre thick is not a correction; it can be the larger term.

The load, second, and in both directions at once.

The crown statistic, third, and smaller on a knit than on a woven cloth because the two diameters already run crown to crown.

The yarn’s own flattening at the crossings, last and smallest, because the force flattening it is a few millinewtons against a woven cloth’s hundreds.

So a knitted thickness measurement is, to first order, a hair measurement — which is the sentence to carry away and the reason the load sweep is the experiment worth running.

Why a standard was written for the other fabric #

The loads in the standard thickness tests were chosen for woven cloth, where the hair layer is a smaller fraction of the whole and the surface has flat stretches for a foot to bridge.

Applied to a knit they are in the wrong regime: a kilopascal is enough to flatten most of a canopy and not enough to move a loop, so the reading is a hair measurement taken at a load chosen to avoid measuring hair. That is not a criticism of the standard, which was written to be reproducible rather than to be structural, but it is a reason not to read a standard thickness as a fabric’s thickness.

The right load for a knit is whatever puts the reading on the flat part of its own sweep, and that is a fabric-by-fabric answer rather than a number in a standard.

Where the ladder goes next #

A thickness that resists a gauge is a thickness that resists a body, and the pressures involved are the subject of what a knit gives up when it is pressed.

And a thickness that holds still air is a thickness that keeps somebody warm, which is a knit is warm because of where its yarn is not.