How thick a knit is

Almost everything this collection computes about a knitted fabric takes a measurement as its input. Munden’s constants give the wale and course spacings; the loop length comes off the machine; the yarn’s diameter comes from its count and a packing factor. The model then says what the fabric does, not what its dimensions are.

The thickness is the exception. It comes out of the geometry with nothing fitted at all, and because it does, it is the one number here a tape measure could contradict outright.

The prediction #

A plain jersey is two yarn diameters thick.

For the 20 tex cotton this collection quotes throughout, whose diameter is 0.167 mm, that is 0.334 mm.

The derivation is three lines. A loop’s feet passed through the head of the loop below, so they are on the far side of it. Two threads in contact have their centre lines one diameter apart. Each centre line has a radius of yarn outside it. One diameter plus two radii is two diameters.

There is no constant in that, no fit, no measurement and no choice.

What it does not depend on #

The striking part is the list of things that do not appear.

Not the loop length. Knit the same yarn at a 5 mm loop instead of 3.5 and the fabric is half again as open, a third lighter, far more extensible and exactly as thick. The interlacing still passes at a diameter.

Not the tightness factor, for the same reason. The one group that collapses everything else about a knitted loop’s geometry has no effect here at all.

Not the fibre, except through the diameter. A wool and a cotton at the same count differ in diameter by eight per cent because their densities differ, and the thickness follows that and nothing more.

Not the machine, on one bed.

So a knitter who wants a thicker fabric out of a given yarn cannot get it by knitting looser. The fabric gets more open, lighter and floppier, and stays as thick as it was. The two levers that do move it are a coarser yarn and a second bed.

Which is not what most people expect #

It is worth dwelling on this because the intuition runs the other way, and the intuition is not stupid.

A loosely knitted fabric feels thicker. It is softer, it compresses further before it resists, it drapes into a heavier-looking fold, and a hand reads all of that as bulk. What it is not is further from face to back at any point where there is yarn.

The two readings come apart because a hand measures a resistance and a gauge measures a distance. The looser fabric has fewer stitches per unit area, so its through-thickness force is spread thinner — from 57 kilopascals at a 2.6 mm loop down to 2.9 at a 5 mm one, a factor of twenty. It gives more easily under a finger and it does not start from anywhere higher.

Why the interlacing is the only term #

It is worth asking why nothing else appears, because a fabric has plenty of other lengths in it.

The wale spacing does not appear because it is a distance along the fabric. The course spacing does not, for the same reason. The loop length does not, because it is consumed by the two spacings. The crimp height — the term that carries most of a woven cloth’s thickness — has no knitted analogue at all, since a knitted loop’s excursion through the fabric is not a wave amplitude but a single monotone climb from one face to the other.

What is left is the one place the fabric has depth: the crossing, where two threads pass. Everything about the loop’s shape happens in the fabric’s plane, and the only thing that happens through it is the interlacing.

That is why the answer is so bare, and it is also why it is so exposed. A model with one term in it has nothing to hide behind.

Where it sits beside a woven cloth #

This collection holds measured thicknesses for eight woven cloths, and they run from 0.16 mm for a voile to 0.44 for a duck.

A 20 tex jersey’s predicted 0.334 mm sits inside that band, between a sheeting and a duck. That is not a strong test — the band is wide and the prediction is a single number — but it is the kind of check worth making before anything else: a model that put a jersey at three millimetres or at thirty micrometres would be wrong in a way no refinement would fix.

The comparison is also unfair in a specific direction. The woven figures are measurements and the knitted one is a prediction, and the measurements are systematically larger than what the same geometric reasoning gives for a woven cloth. This collection has that discrepancy already: Peirce’s circular section over-predicts every measured woven thickness, and by enough that the reason had to be found rather than absorbed.

Why a gauge reads more #

Three reasons, and only one of them is about the model being wrong.

A gauge presses on crowns. A thickness gauge lowers a foot onto the fabric under a stated load, and the first thing it meets is the highest point of the highest loop, not a plane. This collection’s own rung on the subject makes the point in general: a thickness is a maximum, not a mean, and a fabric’s measured thickness is a statistic about its highest crowns.

A gauge measures through the hair layer. Every spun yarn has fibres standing off it, and on a knitted fabric they stand off a surface that is already mostly crowns. The canopy above a fabric is hundreds of micrometres deep on a raised cloth and tens on a smooth one, and a light gauge does not push through it.

And the model’s yarn is a cylinder. A real yarn is compressible and slightly flattened at the crossings, and where it is flattened it is wider than a circle of its own area rather than thicker. That one pushes the measured thickness the other way and is the smallest of the three.

So the prediction is a lower bound on what a gauge will read, and the gap between them is the hair layer plus the crown statistic — both of which this collection computes elsewhere, and neither of which is in the two diameters.

The comparison with Peirce’s cloth, which fails usefully #

There is a temptation to check the knitted prediction against the woven method, and it does not survive contact.

A woven cloth’s thickness in Peirce’s geometry is two thread diameters plus a crimp height, and it over-predicts every measured thickness in this collection’s table — by between fifteen and seventy per cent. The reason is that a woven cloth’s threads flatten at their crossings and the circular section does not know it.

A knitted fabric has the same flattening at its crossings and the same circular section in the model, so the same over-prediction ought to apply. It cannot be checked, because there is no measured knitted thickness here to check it against — and if it does apply, the two diameters are an over-prediction and the gauge’s reading is larger still for the reasons above.

Two effects in opposite directions, neither measured. That is an honest position and it is a reason to run the load sweep below rather than a single comparison.

What a second bed does #

A two-bed fabric’s thickness is the bed gap plus a yarn diameter: the two loop planes, plus a radius on the outside of each.

That is a different kind of number. It is set by the machine rather than by the yarn, so a rib on a three-diameter gap is twice as thick as a jersey of the same yarn and a rib on a five-diameter gap three times. The gauge does not enter, the loop length does not enter, and the fibre only enters through the diameter that scales the gap.

It also means the thickness is the one quantity on the two-bed ladder that a knitter can set directly. Every other number there — the energy, the two forces, the crossing share — follows from the structure and the gap together. The thickness follows from the gap alone.

And a fabric whose thickness has no answer #

One structure in the family is refused, and the refusal is the sharpest thing on this rung.

A tubular fabric — one course on the front bed, the next on the back — is two fabrics made at once. Its two layers are joined at the selvedges and nowhere within a course, so no piece of yarn runs between them and nothing in the structure decides how far apart they lie.

Reporting a jersey’s thickness for it would be reporting a number about one of its layers. Reporting twice that would be assuming the layers touch. So the model returns no thickness for it at all, with the reason attached, and a tube and an interlock balance for different reasons is where that is worked through.

What the thickness is holding up #

A thickness is not much use on its own. What makes this one worth having is that there is now a force across it.

The through-thickness component of the contact force is 7.8 millinewtons a stitch for the jersey, which over the area a stitch occupies is 15 kilopascals. That is what stops the two faces closing, and it is the first point of a compression curve the knitted side of this collection has never had.

The woven side has had one for several rungs: a bearing curve saying how much of a cloth is in contact at a given depth, a thickness that falls under load, and a pressure at every point of it. The knitted side had a plan and no depth. It has a relaxed thickness and an initial slope now, and what a knit gives up when it is pressed is where the two are put beside one another.

The check that is available and has not been run #

This is the most testable claim on the ladder and it deserves to be stated as an experiment rather than as a conclusion.

Take one yarn. Knit three swatches at loop lengths of 3.0, 3.5 and 4.5 millimetres. Relax them all fully — wet, tumble, dry flat. Measure each under a light and stated load.

The model says all three read the same, to within the accuracy of the gauge. If they do not, and in particular if the loosest reads thicker, then the thickness is not set by the interlacing alone and something else — a crown statistic that varies with gauge, a hair layer that varies with the fabric’s openness — is in the measurement.

That is a two-afternoon experiment with no apparatus beyond a gauge, and its outcome would be informative either way.

Why it was not available before #

The reason this collection has not had a knitted thickness until now is not that the arithmetic is hard. It is three lines.

It is that the loop was solved as a plane curve, and a plane curve has no thickness. The interlacing was a point where two centre lines passed a diameter apart across the fabric — a statement made in words, in a model that had no direction to make it in. There was nothing to measure the diameter along.

Once the loop is allowed out of the fabric’s plane, the same sentence becomes arithmetic. That is the general pattern of this whole ladder: the quantities that arrive are not the ones that needed a better calculation, they are the ones that needed a coordinate.

What this does not settle #

A raised or brushed knit. Raising pulls fibre out of the yarn and stands it off the surface, and the resulting fabric’s thickness is mostly hair. The two diameters are still the fabric’s; they are no longer what a gauge is measuring.

A fabric under load. The prediction is the relaxed thickness. Under any real pressure the yarn flattens, and a flattened thread’s geometry is a subject this collection treats at length on the woven side and not at all here.

A spacer fabric. A knitted spacer has a third yarn running between two faces, holding them apart by design. Its thickness is set by that yarn’s length and this model has no term for it.

And what a knitted fabric’s thickness is for. Warmth, chiefly, and a knit is warm because of where its yarn is not is where a thickness becomes a thermal resistance.

What is genuinely new here #

Two things.

A knitted fabric’s thickness, predicted rather than measured, at two yarn diameters on one bed and a bed gap plus a diameter on two.

And a prediction of independence. The thickness does not move with the gauge. That is more falsifiable than the number itself: a single measured thickness can always be explained away by a hair layer or a crown, and three measurements that ought to agree and do not cannot.

What the pictures cannot show #

Both sections here are drawn with the thickness expanded against the width, because a fabric a third of a millimetre thick and four fifths of a millimetre to the wale is otherwise a line with two dots on it. So no proportion in either picture is true, and the numbers in the captions are the only place the scale lives.

Nor does either picture show a hair. Every fabric these sections describe has a fuzz of protruding fibre on both faces, and a gauge meets it before it meets anything drawn here.

What is worth taking away #

Two yarn diameters on one bed, a bed gap plus a diameter on two, and the same number in every relaxation state.

It is the only dimension of a knitted fabric this collection predicts rather than takes as input, which makes it the only one a measurement could contradict — and the prediction that it does not move with the gauge is sharper than the value itself.

Sharper, because a value can be met by accident and an invariance cannot. Any model with a free constant in it can be made to produce 0.334 mm for a 20 tex cotton; producing the same number for the same yarn at every loop length between two and five millimetres, while every other dimension of the fabric changes by a factor of two, is a statement that has to be right for a reason. The reason here is that the thickness is set where two lengths of yarn cross and nowhere else, and the crossing is the one part of the loop that the loop length does not reach.

And it is the prediction that is easiest to test. A mill has no difficulty knitting the same yarn at three settings, and the thicknesses are read on an instrument it already owns. If the three readings differ in the way the courses and wales do, the account here is wrong; if they differ only by what the gauge’s own pressure explains, it is not. That is a day’s work with no special equipment, which is a rarer thing in this subject than it sounds.

Which rungs this stands on #

That the loop’s plane is tilted rather than flat, which is the rotation identity and is what gives the fabric a thickness to have at all.

That the interlacing passes at a diameter, which is not a result but a definition of what a crossing of two threads is — and is the whole of the derivation.

And that a two-bed fabric’s climb is its bed gap, from a rib climbs a gap, which is what turns the thickness into a machine setting.

The three between them are the shortest derivation on this ladder and the most exposed, because a bare prediction with no fitted constant has nothing to absorb a discrepancy.

The one thing that would make it wrong and not obviously so #

If a knitted fabric’s two loop planes were not exactly one diameter apart, everything here would move and nothing would look wrong.

They could fail to be, in one direction. The interlacing is two centre lines passing at a diameter if the two threads are in contact, and a knitted fabric’s crossings are in contact because the loop presses them together — with 7.8 millinewtons through the thickness, which is not much. A fabric whose loops were held apart by anything at all would be thicker.

What could hold them apart is the yarn’s own hair, caught between the two threads at the crossing. On a smooth filament yarn there is none and the prediction should be exact; on a hairy spun yarn there is, and the crossing sits open by some fraction of a fibre diameter.

That is a small effect in the same direction as everything else on this page, which is the awkward part: it makes a measured thickness larger, like the other three, and cannot be separated from them by a single reading. Only a load sweep can, because a fibre caught at a crossing is squeezed out at a pressure the hair layer does not survive either.

Where the ladder goes next #

A thickness with a force across it is a compression curve waiting to be drawn, and the woven half of this collection has been drawing them for several rungs: what a knit gives up when it is pressed is where the two halves are finally compared.

A thickness is also the largest single term in a fabric’s warmth, which is a knit is warm because of where its yarn is not.