Pile is a third thread system

Everything on this site so far has been one encoding. Warp up or warp down at every intersection of two thread systems, over a repeat that tiles the plane — and the insistence has been that this is not a convenient notation for a weave, it is the weave, which is why floats and interlacings and integrity are decidable rather than debatable.

Take a piece of velvet. Two of its thread systems interlace in an ordinary way and make an ordinary matrix. A third stands out of that cloth at right angles, anchored at intervals, forming the surface a hand feels and the eye sees. Where is it in the matrix?

Nowhere. There is no cell for it. A matrix has one entry per crossing of two systems, and with three systems there is no such thing as the crossing.

The site’s own rule, applied to the site #

The invariant this collection has kept from its first essay is that a figure may not claim the machinery went where it did not. The matrix knows nothing about yarn; the trellis knows nothing about stiffness. This is the same rule turned on the encoding itself, and the honest statement is uncomfortable and short: for a pile fabric, the matrix describes the ground and is silent about the pile.

That silence is not a small omission. The pile is why the fabric exists. Nobody buys velvet for its ground cloth, and a carpet’s entire service life is a question about what its pile does. The part of the fabric the encoding handles exactly is the part nobody is interested in.

So this essay is where the field’s argument starts, and the argument is not that the encoding is wrong. It is exact and it stays exact. The argument is about where its edge is, and what — if anything — survives being carried across it.

What survives: the criterion, because it never needed the matrix #

Something does survive, and identifying it is the useful part.

The integrity criterion has been stated on this site in terms of a matrix from the beginning: split the strands into an upper set and a lower set, and the cloth comes apart if at every crossing between them the upper strand is on top. But look at what that actually consumes. It consumes a list of contacts — pairs of strands, with a verdict about which is above — and nothing else. The matrix was a convenient way of generating that list for two systems. It was never the input.

So a pile fabric can be handed to the same criterion. The strands are the ground ends, the ground picks and the pile ends. The contacts are the ground’s ordinary crossings, plus every place the pile passes beneath a pick or lies over one. The above-and-below digraph is built, Tarjan is run on it, and the number of separable pieces comes out — from exactly the same implementation that decides a twill.

And it still rejects #

A criterion that only ever says yes is not a criterion, and this site’s habit is to feed its machinery something it must refuse. The refusal here is exact and it is the failure the criterion was built for in the first place.

Take the pile end and bind it to nothing. Leave it lying over every ground pick, anchored at no point. The picture barely changes — a thread on top of a cloth still looks like a thread on top of a cloth — and the digraph changes completely, because that strand now has no edge running into it from anything. It is its own component. Two pieces.

That is precisely the “warp end lying loose on the surface which can be pulled straight out” that the foundation essay introduced the criterion for. It has been carried into a three-system construction without modification, and it works.

What does not survive #

Three things are lost at the boundary, and they are lost properly rather than approximately.

Float length. A warp float is a run of consecutive picks over which one end stays on the face. A pile end is over the ground everywhere except at its tufts, so its “float” is the whole repeat minus a few points — which is a true statement and a useless one. The quantity that means something for a pile is the tuft spacing, and it is a different quantity with different units.

Interlacings per repeat. The firmness number counts face changes, and a pile end changes face twice per tuft regardless of how long the pile is or how densely it is set. It measures nothing about the pile.

Cover. Cloth cover is the fraction of a plane occupied by threads. A pile does not lie in the plane. The cover of a velvet’s ground is computable and says nothing about the velvet, and a pile fabric’s density is a mass per unit area rather than an area fraction.

Losing three of the site’s principal measurements at one boundary is worth being blunt about. It is not that they give wrong answers for pile fabrics; it is that they answer questions about the ground while appearing to answer questions about the fabric, which is the more dangerous failure of the two.

Which constructions this covers #

“Pile” is one word for a family, and the family divides on two independent questions that are easy to run together.

Which system supplies the pile. A warp pile is fed from a second beam and includes velvet, plush, terry and most carpet. A weft pile is put in by an extra shuttle and includes velveteen and corduroy. The two are made on quite different looms and the arithmetic of their pile heights has nothing in common, which the next two rungs work out.

Whether the pile is cut. A loop pile leaves the pile standing as a closed loop — terry towelling, loop-pile carpet, uncut moquette. A cut pile severs it, so each loop becomes two upright legs. Cutting doubles the number of free ends and halves their length, and it changes the fabric’s behaviour completely while changing the matrix of its ground not at all.

The two questions cross to give four families, and every one of them is a three-system construction with no matrix. What differs is where the pile comes from and what is done to it afterwards.

A terry is the other construction the same argument covers, and its third system is a loop rather than a cut end.

A third system is not automatically outside the matrix #

It would be tidy to say that two systems make a matrix and three do not, and it is not quite true. This site has met a third system before and it stayed inside.

Braids are an interlacement of several strand systems running obliquely, and they were handled at foundation by the same connectivity argument. Backed and stitched cloths carry a second warp or a second weft — genuinely more than two systems — and they do have a matrix, because the extra system is threaded in among the others and crosses them at ordinary intersections. A double cloth is four systems and one matrix.

So the boundary is not the count. What puts a pile outside the encoding is that it leaves the plane. Every system in a backed cloth lies in the same sheet and crosses the others in a grid; a pile end departs from the sheet between its anchor points and is somewhere the grid does not go. There is no intersection to record because the two threads are not in the same place.

That is a more useful statement of the limit than a count of systems, and it predicts the rest of this field correctly. Leno stays in the plane and is outside the encoding for a different reason — its ends change order — which is why the two constructions need different accounts and why neither is a special case of the other.

Periodicity, which the pile has and the matrix cannot hold #

A pile fabric is periodic in a way worth separating from its ground’s repeat, because the two are independent and get conflated.

The ground has a repeat: so many ends by so many picks, in the ordinary sense, and it tiles the cloth. The pile has a tuft spacing, which is how many picks pass between one anchor point and the next along a pile end, and a tuft pitch across the cloth, which is how many warp positions apart the pile ends sit.

Neither is a property of the ground’s repeat and neither divides it in any necessary way. A velvet with a plain ground on a two-by-two repeat may tuft every third pick, and the combined structure then has a period of six picks that appears nowhere in either description on its own. Carpet specifications quote these separately for good reason — rows per inch and pitch are the two numbers on every carpet, and neither is the ground weave.

So the loss at this boundary is not only that the pile has no cell in the matrix. It is that the fabric has a periodicity the matrix’s repeat does not describe, and quoting the ground’s repeat as the fabric’s repeat is wrong rather than incomplete.

The two periods decide how many kinds of tuft there are #

The independence of the ground’s repeat from the tuft spacing has a consequence the section above stops just short of, and it is arithmetic rather than observation.

Write the ground’s repeat as r picks and the tuft spacing as s picks. The combined structure repeats after their lowest common multiple, which is where the six picks of the velvet example came from — a plain ground at r = 2 tufted every s = 3 picks.

Now ask where the tufts land within the ground’s repeat. Over one combined repeat there are lcm(r,s) ÷ s tufts, and that quantity is r ÷ gcd(r,s). Each of them sits at a different pick number modulo r, and each such position occurs exactly once.

So a pile fabric contains r ÷ gcd(r,s) distinct kinds of tuft, in equal numbers.

For the velvet above that is two: half the tufts anchored under a pick where the ground end above them is raised, half where it is lowered. The fabric looks entirely uniform and its anchorages are not.

Which is a property the criterion cannot see #

Every one of those classes is connected, and the criterion certifies every one of them in identical terms — which is the same silence the section below is about, arriving from a second direction.

The classes differ in what is actually holding the tuft down. A tuft bound at a pick that is itself held tightly by the ground sits in a different mechanical situation from one bound at a pick the ground has left slack, and the firmness of the ground at that point is exactly what varies with the phase. The topology is the same and the grip is not.

And the weakest class sets the shedding rate, because a carpet loses the tufts that come out first rather than the average tuft. A fabric with two classes in equal numbers and a two-to-one difference in withdrawal force does not behave like one with the mean force throughout; it behaves like the weaker half, with the stronger half still in place afterwards.

That is the quantity the next rung supplies, and this is the reason it has to be supplied per class rather than once.

And it gives the designer an integer condition #

The condition for a single class is immediate. There is one kind of tuft exactly when r ÷ gcd(r,s) is one, which is to say when the tuft spacing is a multiple of the ground repeat.

That is a strong constraint and it costs something. The smallest multiple is s = r, which puts a tuft in every ground repeat — the densest pile the ground can carry. Any sparser pile means a larger multiple, and the multiples are spaced r picks apart, so a designer wanting a pile between two of them must either accept a mixed population or change the ground.

So uniform anchorage and sparse pile pull against each other, and the pull is arithmetic rather than economic. Three-pick terry on a plain ground — a standard construction — has r = 2 and s = 3, so it has two classes and cannot avoid them without moving to four-pick terry or a four-end ground.

Neither is a defect. It is a fact about the construction that the matrix cannot state, the criterion cannot see, and the loom produces whether or not anyone has noticed.

The gap the criterion leaves #

One thing has been established and one thing has been carefully not established.

Established: the pile is attached, and the criterion says so exactly, and it says the opposite exactly when the pile is not attached.

Not established: how well. A tuft bound under one pick and a tuft bound under three are both connected, both give one component, and both are certified by the criterion in identical terms. One of them is a carpet specified for a hotel corridor and the other sheds its pile in a year.

That gap is not a defect in the implementation and cannot be closed by improving it. The criterion is topological — it asks whether a separation exists, and a separation either exists or does not. Whether a tuft comes out under load is a question about friction, and friction is not a topological property. The criterion is not weak here; it is answering a different question from the one a carpet buyer is asking.

The next rung is entirely about closing that gap with a second model, and about keeping the two apart while doing so.

Why this is the interesting boundary rather than an awkward one #

It would be possible to treat all of this as an inconvenience — a family of fabrics the site’s machinery happens not to cover — and move on. That would miss what the boundary is worth.

An encoding that describes everything explains nothing. The reason the matrix has been such a productive object here is that it is narrow: it commits to two systems in a plane crossing at every intersection, and because it commits, a great many questions about cloth turn into decidable questions about a small integer array. The price of that is a definite edge, and finding the edge is how the commitment gets stated precisely rather than assumed.

Three things have come out of walking up to it in this essay, and none of them would have been visible from inside.

The criterion never depended on the matrix. That was invisible while every construction had one; it becomes a fact about the criterion the moment something does not.

The measurements did depend on it, and differently from one another. Float length degenerates into a true and useless statement, interlacings measure something real about the wrong system, and cover measures the ground while appearing to measure the fabric. Three different failure modes from one boundary.

And the boundary itself turned out not to be about the number of systems, which was the obvious guess and is wrong.

What was counted, and how #

The construction is built as a named graph rather than as a matrix. Ground ends and ground picks get an edge each way according to the ordinary rule — where the end is on the face, the pick is below it, so an edge runs from pick to end. The pile end gets an edge to each pick it passes beneath and an edge from each pick it lies over.

That named graph is then translated into the indexed adjacency lists stronglyConnected consumes, and the direction convention is the one the weave code uses and must stay so: an edge runs from the lower strand to the upper. This is worth stating because reversing it produces a graph that is strongly connected exactly as often, for entirely different reasons, and there is no output that would reveal the error.

Two assertions run. The construction with tufts is asserted to be one cloth; the construction without is asserted to be two. The second is the one doing work — a criterion that certifies everything certifies nothing — and it is written as an assertion rather than a remark for exactly that reason.

Where the model stops #

The ground is drawn as a plain weave and need not be. Real pile fabrics use whatever ground suits, and the criterion would handle any of them; plain is chosen here because it is the clearest thing to draw and nothing in the argument depends on it.

One pile end stands for many. The graph carries a single pile strand, which is enough to decide attachment and says nothing about pile density, distribution or the pattern a Wilton carpet makes by bringing different coloured pile ends to the surface in different places. That last is a whole subject and it is a subject about which the criterion has nothing to say.

Nothing here is about the loom. Velvet is woven face to face and cut apart, terry needs two beams, corduroy needs a cutting machine after the loom. Those are the constructions’ real difficulty and this essay is about their structure.

Who found it, and when #

Pile fabrics are ancient — pile carpet knotting predates any written account of it, and cut velvet was a mature Italian industry by the fifteenth century — and none of it was arrived at through anything resembling this analysis.

What is modern is the framing. Describing a fabric as a set of thread systems with an incidence structure, and asking which properties are decidable from that structure, is a twentieth-century habit that arrives with the mathematics rather than with the craft. It is the same move that turns a knot into a diagram and a diagram into a polynomial, and its value is the same: it makes the boundary of what has been described visible, which is what the rest of this field is about.

Where the ladder goes next #

The criterion has certified that the pile is attached and refused to say how firmly. The next rung supplies the missing quantity from a second model entirely — how a tuft is held — and the two rungs after that work out the two pile geometries, which turn out to share no parameter at all: corduroy’s height is half a cut float and terry’s is half an excess let-off.