The curve that says what a cloth touches with

A surface is not rough by an amount. It is rough in a shape, and the shape that decides what happens when something is pressed against it is the distribution of its heights near the top — because the top is all that participates. Everything below the first few micrometres of a cloth’s crowns is out of contact and stays out of contact until whatever is pressing has crushed its way down to it.

The claim #

The bearing-area curve of a woven cloth opens as the square root of the depth if the weave has a float anywhere, and in proportion to the depth if it does not.

Two things follow and the second is the surprising one.

Two curves with different exponents diverge without limit. The lighter the touch, the greater the difference between a satin and a calico — not asymptotically the same with a constant between them, but further and further apart as the load falls towards nothing.

And the plain weave is the only draft in the four-by-four catalogue on the wrong side of the divide. A weave every one of whose floats is one crossing long is an alternation in both directions, and the only such drafts are the plain weave’s own two translations. Every other cloth in the catalogue touches along lines.

The construction, which is borrowed #

The curve is Abbott and Firestone’s, published in 1933 for the surfaces of machined metal, and it has been the standard description of a bearing surface ever since. It is not a textile idea and this collection does not claim it. The bearing-area curve is the cumulative distribution of surface height, read from the top down: at a depth δ below the highest point of the surface, it reports what fraction of the plan area lies at or above that level.

Its usefulness is that it converts a question about a surface into a question about a number. What area is a plate touching? The bearing area at the depth it has sunk to. What area is a wire tooth catching? The same. What is a coating filling? The complement of the same. Every one of them was previously a matter of judgement about a picture.

Applied to a woven fabric it is being asked of a surface computed from the draft rather than measured off a specimen, so the whole of it is available exactly, at any depth, for any weave, before anything has been woven.

The argument, and it is one paragraph #

Consider the very top of a cloth, where the depth is a fraction of a thread’s radius.

Where a thread floats, its crown is a horizontal line, and a cylinder of radius r cut at a depth δ below its top presents a strip of width 2√(2rδ − δ²). The bearing area is that width times the total length of crown line in the repeat, so at small δ it goes as √δ.

Where no thread floats, every crown is the top of a turn: curved across the thread by its own radius and along the thread by the arc of radius D/2 it is turning through. A doubly curved summit cut at depth δ presents an ellipse of area 2πδ√(R₁R₂), which is linear in δ. So the bearing area goes as δ.

That is the whole derivation. The exponent is not a fitted slope or an asymptotic estimate; it is the difference between cutting a cylinder and cutting a dome, and which one a cloth offers is decided by whether any thread in it is on the face for two crossings running.

What was counted, and how #

Twice, by two routes that share nothing.

In closed form, from the float map: the total length of horizontal crown line per repeat is the sum over every face run of (length − 1) spacings, and the number of point crowns is the count of face runs of length one. Multiply by the strip width or the ellipse area and the answer follows with no sampling anywhere.

By counting, on the sampled surface: the field is evaluated on a grid, cut at a depth, and the samples above the cut are counted.

They agree to about four per cent on a plain weave and one on a twill at a depth of four thousandths of the cloth’s thickness, and the residual is understood — a sample cell whose centre lies above the cut is counted whole, which over-counts by roughly one cell across a strip only a few cells wide. Neither route is a check of the other’s arithmetic; they are a check that the closed form is describing the surface the sampler actually built.

What the exponent is worth #

At a depth of a hundredth of the cloth’s thickness — about four micrometres on an ordinary sheeting — the eight-end satin is touching 11.1 per cent of the plan and the plain weave 1.15. That is a factor of ten, which is interesting but not remarkable.

The remarkable part is what happens as the touch gets lighter. Halve the depth and the satin loses a factor of √2 while the plain weave loses a factor of 2, so the ratio grows by √2. Halve it again and it grows again. There is no depth at which the two curves become parallel, because they are not parallel anywhere: one has a slope of a half on log axes and the other has a slope of one, and lines of different slope diverge.

The arithmetic of the divergence is worth writing out, because it is what makes the claim more than a curiosity. Write the satin’s bearing area as A_s = a√δ and the plain weave’s as A_p = bδ. Their ratio is (a/b)·δ^(−1/2), which has no limit as δ falls. Putting the sheeting’s own numbers in, as a fraction of the cloth’s thickness: the ratio is 6.7 at two hundredths, 9.6 at one hundredth, 13.7 at a two-hundredth, 21.7 at a five-hundredth, 30.7 at a thousandth and 68.8 at five thousandths. Nothing about the cloth has changed at any point; only how hard something is being pressed against it.

The practical consequence is that the lighter the contact, the more the draft matters. A cloth under a heavy press is a cloth whose crowns have all merged and whose bearing area is nearly its cover factor, and at that point the weave has stopped mattering. A cloth under a finger, a cloth against a skin, a cloth in the first instant of a rubbing cycle — those are the regimes in which a satin and a calico of the same yarn behave like different materials.

One system bears, and it is not the one on the face #

The curve has a second feature at its very top, and it is easy to miss because it is only a few micrometres tall.

A cloth’s two systems crown at different heights whenever their crimps do not divide in proportion to their diameters, which is nearly always. So the first thing anything touches is the higher system alone, and the lower one is not in contact at all until the approach has passed the step between them. In an ordinary sheeting that step is eight micrometres — two per cent of the cloth’s thickness, and four times the depth a thickness gauge reaches.

Which means the top of the bearing curve belongs to one thread system, and the curve has a visible knee where the other one arrives.

The consequence for wear is direct and is taken up later in this ladder: whatever rubs a cloth rubs one system for the first several micrometres and both after that, and which system that is depends on a division of crimp that nobody actually measures.

Only two drafts in the whole catalogue #

The divide has an exact census on the other side of it, and the answer is small.

Of the 22,874 four-by-four drafts in which every end and every pick interlaces at least once — the same catalogue every enumeration on this site runs over — exactly two have no float anywhere. They are the plain weave and its translation by one thread, which are the same cloth written two ways.

So the exception is not a family, it is a weave. That is a stronger statement than it looks, because the plain weave is also the exception in the size of hole a cloth has — the only draft whose apertures are all the same — and it is the exception for the same structural reason both times: it is the only draft with no run of two anywhere, so it is the only draft with no second scale in it. Two properties of a cloth that have nothing to do with one another single out the same two matrices, because both are consequences of the same absence.

Where the model stops #

The curve is geometry and contains no force. It says what area lies within a depth; it does not say what pressure is needed to sink that far, and turning one into the other needs a transverse stiffness that this collection has shown has no lower bound at all. Every pressure in this ladder is quoted with the fitted value it came from named beside it.

The hairs get there first. What actually meets another surface is a layer of fibre ends standing off the yarn, and against them the bearing area computed here is a lower bound. On a smooth filament cloth the difference is small; on a soft-spun staple yarn it is not, and on a raised cloth the whole construction is describing the wrong surface.

The plateau is taken as flat and a real float sags between the threads it rests on. That rounds the top of a ribbon slightly, so the true opening near zero depth is a little slower than a pure square root — but it does not shorten the ribbon, and the ribbon’s length is what the exponent argument turns on.

The two systems are treated as one surface, and for the bearing curve they are not. The construction above takes the maximum of the two threads’ heights at every point, which is right, but it then reads a single curve off the result — and the top of that curve belongs to one system while the rest of it belongs to both. Everything said here about exponents applies to the higher system’s crowns; the lower system’s contribution enters as a second copy of the same argument, offset by the step, and a cloth whose two crowns happen to coincide has both copies arriving at once.

And nothing here is worn or washed. A cloth that has been used has crowns that have been flattened, which is the same thing a calender does deliberately and which moves the curve enormously. That is what the finish buys, computed in its own place.

Where the knee is, and why the fitting range is not a choice #

The step between the two systems’ crowns is quoted at eight micrometres and the exponent is fitted “over the top two per cent of the thickness”, which is the same distance. That coincidence is not one, and noticing it says what the fit’s upper limit has to be.

The step follows from the crimp division. With the two crimp heights summing to the combined diameter D and dividing in a ratio r, the crowns differ in height by

D(r − 1) ÷ 2(r + 1),

which for the sheeting’s 0.334-millimetre combined diameter gives eight micrometres at a crimp ratio of 1.10 — a very slightly unbalanced cloth, which is what a sheeting is.

Above that depth both systems are in contact and the bearing area jumps. A power law fitted across the jump reads the step as part of the curve and reports a slope that is too steep, so a fit that runs past the knee over-states the exponent for every weave.

So the fitting range is not a methodological choice; its upper limit is the crimp step, and the essay’s two per cent is right because it stops there.

That is worth stating because the crimp ratio is a quantity nobody measures — so the range over which an exponent measurement is valid depends on an unmeasured number, and a laboratory fitting over a range chosen for convenience has no way to know whether it crossed a knee.

What the crimp ratio does and does not touch #

The same expression separates the crimp ratio’s two effects, and the separation is the reassuring one.

It sets the coefficient. A cloth whose two crowns coincide has both systems bearing from zero depth, so its bearing area is roughly doubled at every depth — a factor of two in the coefficient.

It does not touch the exponent. Both systems open as square roots if they have floats and linearly if they do not, so adding a second copy of the same power law changes the multiplier and leaves the slope where it was.

So the essay’s central claim — that the exponent is a half or a one according to whether the weave has a float — is untouched by the crimp division, which is the input this collection is least confident about. Everything the crimp ratio moves is a coefficient and everything the draft decides is an exponent, and the two are the quantities this ladder wants to be robust and is willing to lose respectively.

Which gives the balanced cloth a second distinction #

The formula also says which cloths have a knee at all, and the answer picks out a familiar case.

A balanced cloth in equal yarns has a crimp ratio of one and no step whatever. Its two systems crown at the same height, both are in contact from zero depth, and its bearing curve is a single clean power law all the way down with no knee to fit around.

So a balanced cloth touches twice as much as an unbalanced one of the same yarn at every shallow depth, and its exponent can be fitted over any range short of crown merging rather than over the top two per cent.

That is a third property on which balance is the special case, after it forcing integrity and it minimising crown line — and here it is the case where a measurement is easiest rather than where a structure is safest.

The unbalanced cloths are the ones whose surfaces are hardest to characterise, because their curve has a feature in it whose position depends on the one geometric quantity nobody has measured.

The generalisation #

When two rough bodies meet, the geometry near the top of each decides the contact, and the top of a surface is characterised by an exponent rather than by an amplitude.

The transferable form is that a surface built from lines and a surface built from points are different kinds of surface, and no measurement of roughness distinguishes them. A root-mean-square height, a peak-to-valley, an arithmetic average — all three of them come out very nearly equal for the satin and the plain weave in these figures, and the two cloths touch the world with contact areas that differ by any factor one cares to name simply by choosing a light enough load.

That is a warning about roughness numbers in general. They are single moments of a distribution whose shape is the thing being asked about, and a single moment of a distribution cannot tell a ridge from a summit.

What it does to three older results #

The curve does not replace anything already on this site, but it reframes three things that were separate.

The thickness that is a maximum. A presser foot rests on whatever is highest beneath it, so a measured thickness is an extreme value and grows with the area of the foot. The bearing curve is the other half of that statement: the extreme decides where the foot starts, and the curve decides how far it then sinks. Together they say a thickness measurement has a population effect and a geometry effect in it, and neither is in the number the standard reports.

Abrasion, which is two quantities. A satin does not wear quickly but fails badly — the argument being that a flat face spreads the rubbing over more thread than a plain weave’s crowns do. The bearing curve puts a number on the spreading and it is far larger than that essay could have supposed: not a factor of two but a factor that grows without limit as the load falls.

And the jam. A cloth jams where its threads are thickest, which is an extreme of the diameter distribution across the plan. The bearing curve is an extreme of the height distribution over the same plan, and both are cases of the same habit — a cloth is decided by the ends of a distribution far more often than by its middle.

Who found it, and when #

Abbott and Firestone gave the curve in 1933 and Abbott’s name is still on it in every metrology standard. The idea that the small-scale form of a contact decides its area, rather than the nominal geometry, is Bowden and Tabor’s from the 1930s and 1940s, and belongs to the physics of friction rather than to textiles.

What this collection contributes is the observation that for a woven fabric the curve is not measured but derived, that it is derived from the draft, and that the derivation produces an exponent rather than a coefficient — so a difference between weaves that everybody knows qualitatively turns out to be a difference of kind that can be written down in one line.

Where the ladder goes next #

To the census: two drafts of twenty-two thousand puts the whole catalogue on one axis and finds where the crown line is largest, which is not where the longest float is.

Then to what the curve is actually used for. How much of a cloth is touching turns the depth into a pressure and finds the answer is a fraction of a per cent; a thickness gauge reads the draft finds the same curve inside a measurement nobody thinks of as a weave measurement; and a cloth loses its strength before its mass reads it as the thing that decides where wear lands.