A cloth has an outside

Almost everything anybody does to a fabric happens at its outside. A presser foot rests on it, a wire tooth catches it, a squeegee drags a film across it, a finger runs over it, and the rubbing that eventually destroys it never reaches further in than the radius of one thread. Yet every number this collection has computed so far is a property of the interior: a crimp is a length, a cover factor is a projected area, a hole is a gap between four threads, a fibre count is a division. Not one of them says where the cloth stops.

The claim #

A woven cloth’s outside is a height field, it is computed from the draft and the Peirce solution with nothing added but one geometric statement, and it has two distinct crown levels rather than one.

Three things follow, and each of them is a fact about a fabric that no quantity already on this site could express.

A float’s crown is a line. Where a thread is on the face over more than one crossing it rests on the threads it passes over, all of which are at the same height, so it is straight and horizontal from the first of them to the last. Its highest points are not a point but a segment.

A plain weave’s crown is a point. With no float anywhere, every crown is the top of a turn, curved along the thread by the crimp and across it by the thread’s own radius. It is a summit rather than a ridge.

And the warp and the weft do not crown at the same height. The top of a warp end stands at h₁/2 + d₁/2 above the mid-plane of the cloth and the top of a weft pick at h₂/2 + d₂/2, while Peirce’s closure condition says h₁ + h₂ = d₁ + d₂. The two agree exactly when the crimps divide in proportion to the diameters, and in no other case.

What the field is made of #

Two constructions, and only one of them is new.

The transition is Peirce’s, unchanged. A thread changing face travels from the top of the cloth to the bottom across one spacing of the other system: an arc of radius D/2 around the thread it has been riding, a straight run inclined at the weave angle, and the mirror arc into the thread it passes under. Those are exactly the two equations the site’s thread geometry already solves — p₂ = (l₁ − Dθ₁)cos θ₁ + D sin θ₁ and h₁ = (l₁ − Dθ₁)sin θ₁ + D(1 − cos θ₁) — read as a path rather than as a pair of numbers to be extracted from it.

The float rests. That is the added statement, and it is one sentence: a warp end floating over four picks lies on four wefts, so it is straight and horizontal over the three spacings between the first and the last of them.

That is not the only way a float can be handed to Peirce, and the difference matters. When this collection asks how much thread a repeat holds — the take-up, which decides what a loom must be set to — a float is given to the same equations as a longer bending pitch: a thread that turns once in four crossings is treated as turning across four spacings instead of one. That is the right question answered correctly, and it produces a gentler, longer bend. It is the wrong construction for a surface, because it puts the thread’s highest point at the top of a single long arc when the thread is in fact lying on four supports. The two models are the same equations at two spacings, they agree exactly where the float is one crossing long, and they answer two different questions about the same cloth. Saying which is being used is not pedantry here: it decides whether the crown of a satin is a ridge or a summit, and everything downstream turns on that.

Across the thread the profile is the thread’s own section, which on this site is a circle or Kemp’s racetrack depending on how hard the cloth has been pressed. The surface of the cloth at any point of the plan is then simply the higher of the two threads that cover it, and where neither covers it there is a hole and no surface at all.

The step that is not the step already computed #

This site already has a step in a cloth’s surface, and it is a different one. A figured cloth has two weaves in one set of threads at one sett, the region that interlaces less often is pressed less often, and it stands about fifty micrometres proud of its ground. That is a step between two weaves.

What is computed here is the surface within one weave, where the crimp is uniform and there is no such step, and it still has two levels — because the warp and the weft are different threads doing different amounts of bending. In a balanced sheeting the difference is small, eight micrometres on a cloth three hundred and eighty thick, but it is not zero and it is not a rounding error: until anything pressing on the cloth has sunk eight micrometres, it is touching one system and not the other at all.

Which makes the most basic question about a cloth’s surface depend on the one number the model leaves free. The crimp ratio is not a measurement — that essay is the whole argument, and its conclusion was that the number every fabric analysis quotes is a bookkeeping convention rather than something anybody weighs. It now turns out to decide which of a cloth’s two thread systems meets the world, which is to say which one wears out.

What was counted, and how #

The height field is built twice over, and the two routes are compared rather than blended.

Once by construction. For each end and each pick, the runs of the draft are found cyclically, the plateaux and transitions are laid down in order, and the lengths are asserted to tile the repeat exactly. An off-by-one here would shift every crown in the cloth by half a spacing and nothing about the resulting picture would look wrong, so the tiling is checked to a part in 10⁹ rather than assumed.

Once by sampling. The finished surface is evaluated on a grid over one repeat — typically sixteen samples per thread spacing in each direction — and every quantity the following essays use is counted off that grid.

The join to the rest of the collection is a single assertion and it is the one that would catch the whole construction being wrong. At a float of one there is no plateau, the path is Peirce’s transition repeated and nothing else, and the crown height must be the one the site’s own solver already reports. Over all eight cloths in the table it agrees to zero — not to a tolerance, to zero, because the same arithmetic is being asked the same question by two different routes.

The two crowns, drawn #

A satin and a plain weave in the same cloth — same counts, same setts, same thickness, same yarn — have surfaces that do not resemble each other in any respect except their amplitude.

The numbers behind the two pictures say the same thing. In sheeting at twenty-eight ends and twenty-six picks to the centimetre, the eight-end satin carries about two millimetres of horizontal crown line in every square millimetre of cloth and the plain weave carries none at all; the two-and-two twill between them carries seven tenths of a millimetre. Their peak-to-valley heights are within a few per cent of one another throughout, because that quantity is pinned by the closure condition and the closure condition does not know which draft it is being asked about.

The amplitudes are nearly equal and the topographies are unrelated. That is the whole reason a surface deserves a description of its own rather than a single roughness number: a cloth’s outside is not rough by an amount, it is rough in a shape, and the shape is the draft.

The face is an area and the draft is a count #

One more quantity falls out of the field immediately, and it corrects a habit this collection has indulged from the beginning.

The draft says which system is on the face at each intersection, and the balance of a weave is the fraction of intersections the warp wins. That is a count. What a reader sees is an area, and the two are not the same number unless the two systems have the same diameter and the same sett.

A warp-faced cloth is less warp-faced than its draft says, whenever the weft is the coarser system — which in a shirting it usually is, because the warp is the finer and denser system by construction. Every statement of the form “three quarters of the face is warp” that this collection has made from a balance figure is a statement about intersections, and the two versions of it differ by several points on an ordinary cloth.

The step between the crowns, in one line #

The two crown heights are said above to agree “when the crimps divide in proportion to the diameters”, which is nearly right and is worth doing exactly, because the exact version is a single subtraction and it names the quantity a designer would want.

The warp’s crown stands at (h₁ + d₁)/2 above the mid-plane and the weft’s at (h₂ + d₂)/2. Peirce’s closure condition pins h₁ + h₂ = d₁ + d₂. Substitute and everything cancels but two terms:

step = crown₁ − crown₂ = h₁ − d₂.

The warp stands above the weft by its own crimp height less the weft’s diameter, and by nothing else. No angle, no sett, no thread length survives the subtraction.

Three readings, and each is a fact the essay’s figures show and its text does not state.

The crowns are level exactly when h₁ = d₂, which by the closure condition also makes h₂ = d₁. So each system’s crimp height must equal the other system’s diameter — not its own, and not its own in proportion. For a balanced cloth with equal diameters that is the equal division of the crimp, which is why a sheeting’s step is eight micrometres rather than zero: its division is not quite equal.

A cloth in which the coarser system also does more of the bending has level crowns. Set d₂ > d₁ and level crowns need h₁ > d₁ — the finer warp must carry more crimp height than its own diameter. That is what a poplin does, and it is the case where the crown step is smallest despite the construction being the most unbalanced in the table.

And the sign is readable from a specification. A cloth whose warp crimp height exceeds its weft diameter presents its warp to the world; one where it does not presents its weft. Both quantities are on any fabric analysis — a crimp height comes from the crimp and the thickness, and a diameter comes from the count — so which system a cloth wears out is computable from the specification sheet, and it is not what the balance says.

That last is the practical payoff and it corrects a habit. The balance counts intersections and this counts height, and a cloth can be warp-faced by count and weft-crowned in fact. What touches the world is the higher crown, whatever the draft’s arithmetic says about which system is up more often.

The caution the closed form carries is the one the whole rung carries: h₁ comes from the crimp ratio, and the crimp ratio is not a measurement. So the step is exactly computable from a number nobody weighs, which is the same uncomfortable position the section above reaches by a longer route.

Where the model stops #

Four places, and the first is much the most important.

The hairs are not in it. The outermost fibres of a spun yarn are held by nothing — the radial pressure inside a twisted yarn falls to exactly zero at its surface — so some of them stand off, and what a neighbouring surface actually meets first is a hair layer rather than the yarn. Everything computed here is a surface of yarn. Against the hairs it is a lower bound on the contact area and an upper bound on the pressure, and on a raised or a brushed cloth it is not the operative surface at all.

The crimp is uniform over the repeat. One Peirce solution serves every intersection, which is right within one weave and wrong the moment two weaves share a set of threads. Where that matters it is exactly the figured cloth’s step, computed elsewhere and taken from there rather than recomputed here.

The float is taken as resting on its supports, and a real one sags. A thread lying across four others is not perfectly straight between them; it is a beam on supports with its own tension and its own bending rigidity, and it dips. The dip is small — the site’s own bending arithmetic puts a thread in cloth at the free end of a bracket three hundred wide, which is to say very flexible, but the span is under a millimetre and the tension is not zero. Including it would round the top of every plateau slightly and would not change any ordering, because the plateau’s length is what the results below turn on and sagging does not shorten it.

And nothing here is pressed. The field is the geometry of an unloaded cloth. What it takes to reach any given depth into it needs a transverse stiffness, and this collection has established that a yarn’s transverse stiffness has no lower bound at all. That refusal is not repaired here; it is carried, and every pressure quoted anywhere in this ladder names the fitted value it came from.

The generalisation #

A surface is a distribution rather than a number, and the useful form of that is sharper: whenever a body meets another body, what matters is not how rough either of them is but how their heights are distributed near the top.

The transferable version is that the highest few per cent of a surface does all the work. Everything that touches a cloth touches its crowns; everything that wears a cloth wears its crowns; everything that reflects off a cloth reflects off whichever part of its crowns happens to be flat. A quantity averaged over the whole surface — a mean height, a mean cover, a mean anything — is averaging over the ninety-five per cent of the material that is not participating.

That is why the curve that says what a cloth touches with does not compute a roughness. It computes the cumulative distribution of height, which is the object that answers the question actually being asked.

Who found it, and when #

The pieces are old and the assembly is not.

Peirce’s geometry is from 1937 and is the whole of the transition. The idea that a surface should be described by the cumulative distribution of its heights rather than by a single amplitude is Abbott and Firestone’s, from 1933, and belongs to the metrology of machined metal rather than to textiles; the bearing curve borrows it by name. The observation that a woven surface has two distinct crown levels is implicit in every published fabric section ever drawn and, as far as this collection can tell, has not been used for anything.

What is new here is only the joining: that the draft, which decides where the floats are, and Peirce’s solution, which decides how high the crowns stand, together determine a surface exactly, with one added sentence about a float resting on its supports and nothing else.

Where the ladder goes next #

Straight to the distribution: the curve that says what a cloth touches with reads the height field as a cumulative area and finds that the difference between a satin and a calico is not a factor but an exponent.

Sideways, the same field answers three questions that have never been connected. What reflects off a cloth is the part of its crowns that is flat, which is a float reflecting into a line. What wears off a cloth is the part of its crowns that is high, which is the strength a fabric loses long before its mass. And what a coating has to bury before it can bridge a hole is the volume above the same surface. One object, three readings, and the draft is in all three of them.