A thickness gauge reads the draft

A fabric thickness is the least controversial number in a textile specification. A foot of stated area is brought down onto the cloth at a stated pressure, the gap is read, and the answer is quoted in millimetres. Everything in that sentence except the millimetres turns out to be load-bearing.

The claim #

A measured fabric thickness contains three corrections to the geometric thickness. Two are large and belong to the yarn; the third is small, belongs to the weave, and is systematic.

The extreme, upward. A foot rests on whatever is highest beneath it, so the reading is an order statistic of the diameter distribution and grows with the area of the foot. That is already established here, and at a standard two-thousand-square-millimetre foot over some fourteen thousand crossings it is worth tens of per cent.

The compression, downward. The cloth is squashed. Every fabric in the table measures well below what a round-section geometry predicts — an ordinary sheeting measures 260 micrometres against a predicted 382 — and reconciling the two is what fixed the transverse stiffness this collection uses.

And the sinking, downward, and it depends on the draft. The foot descends until the bearing area supports the pressure. On a plain weave that is 3.9 micrometres; on an eight-end satin it is 0.9. The two cloths, identical in every other respect, report thicknesses three micrometres apart — about one per cent.

Why the third correction exists at all #

The first two are about the yarn and would be there in any construction. The third is a pure consequence of the exponent.

At the standard pressure the foot is looking for a contact area of a few per cent. A weave with plateaux finds it almost immediately, because its bearing curve opens as the square root of depth and a square root is steep at the origin. A plain weave has to be pushed four times as far, because its crowns are points and its curve opens linearly.

So the sinking is inversely related to the crown line, and the crown line is a census over the draft. Nothing about the yarn has changed; the foot has simply found a different kind of surface to rest on.

What was counted, and how #

For each draft the surface is built, its bearing curve is taken in closed form, and the depth is found at which the pressure the model supplies reaches the standard’s kilopascal. The pressure model is the crude one described in the contact essay: the material in a patch compressed by the approach over the thread’s thickness, at the transverse stiffness the compression work fitted.

Across six drafts of one sheeting the readings come out as follows. A one-and-three twill reports 376.0 micrometres; a plain weave 377.7; a two-and-two twill 379.8; a three-and-one twill 380.5; a five-end satin 380.6; an eight-end satin 380.7. The geometric thickness of every one of them is 381.6.

The spread is 4.7 micrometres, which is 1.2 per cent. The stiffness moves every reading together — across its published range from one to ten newtons per square millimetre, a single cloth’s reading moves by about six micrometres — so the ordering is far more robust than the values, and the ordering is exactly the ordering of the crown line.

The one-and-three twill, which is the informative case #

The weakest reading in the table is not the plain weave. It is the one-and-three twill, at 376.0 micrometres — six microns below the geometric thickness and two below the plain weave’s.

That cloth is three-quarters weft float on its face. It looks, and photographs, like a heavily floated fabric. But its warp stands higher, and the warp in a one-and-three twill is on the face at one crossing in four with no run of two anywhere — so the surface the foot meets is a lattice of isolated summits, and it sinks further into it than into a plain weave, because there are fewer summits.

A thickness gauge, in other words, does not read what a cloth looks like. It reads the system that crowns higher, and which system that is comes from the division of the crimp, which is a convention rather than a measurement. That is the same caveat as everywhere else in this ladder and it bites hardest here, because a thickness is a number people compare between fabrics to three significant figures.

Where the three corrections sit relative to one another #

Setting them side by side is the useful part, because they are not comparable in size and are routinely confused.

The extreme is the largest and is about the yarn’s evenness. The expected maximum of fourteen thousand draws from a distribution at fifteen per cent variation sits nearly eighty per cent above the mean. Not all of that reaches the thickness — the highest crossing raises the foot only where it is, and the foot is rigid over a large area, so the effect is a fraction of the extreme rather than the extreme — but it is a tens-of-per-cent effect and it grows with the size of the foot, which is why the standard specifies one.

The compression is next and is about the yarn’s softness. It is worth about a third on an ordinary cloth, it is what the fitted stiffness was fitted to, and it has no lower bound in theory.

The sinking is smallest and is about the weave. One per cent, systematic, and reproducible, because it is geometry rather than a property of any particular specimen.

The pressure exponent, which is a prediction rather than a number #

The sinking has an algebraic form, and it is different for the two kinds of crown.

Write the pressure as G times the strain in the patch times the bearing fraction, which is G(δ/b)·A(δ). For a weave with plateaux the bearing area is proportional to √δ, so the pressure goes as $\delta^{3/2}$ and the sinking as $P^{2/3}$. For a plain weave the bearing area is proportional to δ, so the pressure goes as δ² and the sinking as $P^{1/2}$.

Those are different exponents and they are a prediction about the shape of a compression curve, not about any one thickness. Doubling the pressure sinks a floated cloth by a factor of 1.59 and a plain weave by 1.41; multiply the pressure by ten and the two factors are 4.64 and 3.16.

That is worth having because it is checkable with equipment every textile laboratory owns. A compression curve — thickness against pressure, over two or three decades of load — is a routine measurement, and it is routinely fitted with an empirical power law whose exponent is reported and not explained. The prediction here is that the exponent is a property of the interlacement, that it is two thirds for anything with a float and a half for a plain weave, and that it should be visible at the light end of the curve and should wash out at the heavy end where the crowns have merged.

It is also the point at which this essay’s small correction stops being small. One per cent at a kilopascal is a curiosity. A different exponent over three decades of pressure is a different curve, and it is taken up in its own place.

What it means for a comparison #

One per cent is below the repeatability of a single thickness measurement, which is why it has never shown up. It is not below the repeatability of a comparison, because it is systematic: measure a hundred specimens of each of two weaves and the difference is still there, at the same size and in the same direction.

So a thickness comparison between two weaves of one yarn is biased, and the direction is always the same: the more floated cloth reads thicker. That is the opposite of what a reader might expect from the fact that a floated cloth is usually the softer one, and it is not an error in anybody’s gauge — it is the gauge correctly reporting that the two cloths present different surfaces to it.

The practical form: a specification that pins a thickness to three figures on a fabric whose weave is not also pinned has left about a per cent unspecified, which is the same order as the tolerance such specifications usually carry.

Two cloths that measure the same and are not the same #

The sharpest way to put the finding is as a pair of fabrics that a specification cannot tell apart.

Take a plain weave and a two-and-two twill of the same yarn at the same sett. Their geometric thicknesses are identical to the last figure, because thickness is the sum of the two diameters and neither the draft nor the sett appears in that sum. Their measured thicknesses differ by 2.1 micrometres, which no laboratory would report as a difference.

Now press harder. At a hundred kilopascals the plain weave has sunk 27.8 micrometres and the twill considerably less, and the gap between the readings has grown by more than an order of magnitude — because the two are sinking along curves with different exponents. The two cloths are indistinguishable by thickness at the standard’s pressure and clearly distinguishable at a higher one, and nothing about either fabric has changed.

That is an argument for measuring compression rather than thickness whenever the question is about a surface, and it is also a warning about the inverse. A quoted thickness carries no information about the weave; a quoted compression curve carries a great deal, and almost nobody quotes one.

Where the model stops #

The foot is rigid and the cloth is not supported. A real gauge presses cloth against an anvil, so the cloth is being compressed between two surfaces and both of them have bearing curves — except that the anvil’s is flat, which is the case computed here. Where two cloths meet the arithmetic is different again.

The pressure model is a bed of springs. It has no proper contact mechanics in it and would move the numbers by tens of per cent. It would not move the ordering, because it enters every draft identically.

The extreme correction is not computed here, only referenced. Combining it properly with the bearing curve means asking for the bearing curve of a surface whose crown heights are themselves a distribution, which needs a joint model this collection does not have. The two corrections are estimated separately and their sizes compared; they are not added.

And the hairs, again. At depths of one to four micrometres, what a foot meets on a spun-yarn cloth is fibre ends rather than yarn. The whole of this essay is an argument about a surface that on a hairy cloth is beneath the one being measured, and the hair layer is a spinning property that would enter every reading before any of this does.

What the foot’s area is actually for #

The standard fixes three things and each of them is fixing a different one of the corrections above, which is worth noticing because it is not how the standard explains itself.

The pressure fixes the compression and the sinking. Both of them scale with it, one to the two-thirds power and one to the one-half, and neither has any meaning without it stated.

The area fixes the extreme. A foot covering fourteen thousand crossings finds a higher maximum than one covering fourteen hundred, so the reading grows with the foot and the standard has to say which foot. That is an extreme-value finding and it is the reason the area is in the standard at all.

And the dwell time fixes nothing that is computed here. A cloth under a foot goes on thinning for as long as the foot is there, because the fibres creep, and the standard’s stated dwell is a decision about where on a relaxation curve to read. That is a rate and there is no rate anywhere in this collection’s compression arithmetic — the site’s standing position is that time enters as a count of events and never as a rate, which is honest and is also an admission.

So of the four numbers a thickness standard specifies, three are pinning down effects that can be computed and one is pinning down an effect that cannot.

The anvil sinks too, and that changes the ordering #

The limits section notes that a gauge presses cloth against an anvil and that the anvil’s own bearing curve is flat. It is — and the cloth’s is not, on that side either.

The underside of a fabric has crowns exactly as the upper side does, and a flat anvil descends into them by exactly the same argument. So the reading is the geometric thickness less two sinkings, one at each face, and the two are the sinkings of a cloth and of its own complement.

That has an immediate consequence which is stronger than a correction:

a thickness gauge cannot distinguish a cloth from its own back.

The sum of a draft’s sinking and its complement’s is unchanged by turning the cloth over, because turning it over exchanges the two terms. So a three-one twill and a one-three twill — the same cloth, two ways up — must read identically, and the single-face arithmetic above has them 4.5 micrometres apart.

That is a check the essay’s own table fails, and it fails in a direction that identifies the missing term rather than the missing number.

What the two-sided readings come to #

Adding each face to its complement, on the same six drafts:

The spread grows slightly — six micrometres against 4.7 — and the ordering changes. The three-one twill, which read second-thickest on one face, now reads below the two-two twill, because its back is a lattice of isolated warp summits and the anvil sinks a long way into it.

So the correction is not a refinement of a small number; it reorders the table. A cloth is thick to a gauge when both its faces have plateaux, which is a property of the pair rather than of the face anybody is looking at — and the balanced weaves, whose two faces are alike, do best or worst together while the unbalanced ones average.

Which makes the gauge read a different quantity again #

Stated as a quantity rather than a correction, the two-sided version says the gauge is measuring the sum of a draft’s crown line and its complement’s — and that sum has a familiar property.

Complementing turns warp floats into weft floats of the same length, so a draft and its complement have the same longest float and the same interlacing count. What they do not share is which system carries the plateaux, and the sum counts both. So a thickness gauge is reading a quantity invariant under complementation, which is exactly the class of quantity a damask’s figure and ground share.

That is a small and tidy result with a use. A damask measures one thickness across its figure and its ground, exactly, because the two regions are complements — where a naive single-face reading would have them differing by whatever the crown lines differ by. The essay on damask says figure and ground agree on every structural quantity; this adds that they agree on the measured thickness too, and for a reason the single-face model would have denied.

And it says which comparison a gauge cannot make. Two cloths differing only in which face is up are one reading, so a specification that pins a thickness has said nothing at all about which side of the fabric is the face — which is a piece of information the trade cares about a great deal and which the instrument was never going to supply.

The generalisation #

A measurement of a boundary is a measurement of a boundary under a stated interrogation, and the interrogation is in the answer.

The transferable form is that any measurement which finds a surface by pressing against it returns a value that depends on the surface’s height distribution near the top — so two objects with identical nominal dimensions and different surface statistics measure differently, systematically, at any stated probing force.

That is familiar in metrology, where a stylus radius and a filter cutoff are always quoted with a roughness figure. It is not familiar in textiles, where a thickness is quoted as though it were a dimension. The nearest thing this collection has to a moral is that a fabric has no thickness: it has a family of thicknesses indexed by the pressure asked, and the standard picks one member of the family and does not say that it has done so.

Who found it, and when #

The extreme-value half is Peirce’s territory, made explicit here by the essay that priced a fabric’s thickness as an order statistic. The compression half is the reconciliation that fitted a transverse stiffness to eight measured thicknesses. The observation that the draft is inside a thickness reading appears to be new, and it is small enough that it could only have been found by computing rather than by measuring: a one per cent systematic difference between weaves is invisible against the specimen-to-specimen scatter of any real fabric.

The standards themselves are careful about the two things they can see. ISO 5084 fixes the pressure and the foot area and the dwell time, and its own notes say that thickness is not a property of a fabric alone. What no standard says is that the weave enters at all.

Where the ladder goes next #

Into the mechanics: a cloth compresses along its own bearing curve, which takes the same curve as a rate rather than a state and explains the shape of a pressure–thickness relation that has always been fitted empirically.

Sideways, the same reasoning applies to every instrument that finds a fabric by pressing on it — a drape meter’s clamp, a stiffness tester’s platen, a wear tester’s abradant — and to the one that finds it by rubbing, where the concentration on the crowns is the whole of the damage.