Velvet is cut apart

This site’s central check has been used two ways since its foundation. Forwards it verifies that a cloth is one cloth. Backwards it verifies that a double cloth is genuinely two, which is what makes the check a measurement rather than a filter.

There is a third use, and it is the way velvet is actually made.

The construction #

Two complete ground cloths are woven at once, one above the other, with a gap between them held open by the loom’s shedding. Each has its own warp and its own weft and neither touches the other.

That last clause is worth taking literally, because it is what makes the arithmetic below work. The two grounds are not a double cloth in the sense an earlier essay here built: there is no stitching, no exchange of face and back, no intersection at which a thread from one meets a thread from the other. Considered on their own they are two fabrics that happen to be in the same loom, and the criterion applied to them alone returns two, immediately and for the most boring possible reason.

A separate pile warp, on its own beam, is drawn into the upper ground for a few picks, then down across the gap and into the lower ground for a few picks, then back up. It shuttles.

A knife — a fixed blade in the older machines, a travelling one in most modern ones — runs down the gap as the cloth advances, severing every pile end at its midpoint. Above the knife there is one fabric; below it there are two, each with a cut pile standing where the shuttling used to be.

What the criterion says, and why it is the whole point #

Run the above-and-below graph on the uncut construction. The upper ground’s ends and picks form a strongly connected set among themselves; so do the lower ground’s. Between the two sets there is not one contact — they are separate cloths in every sense — except through the pile ends, which have contacts in both.

So the whole thing is one strongly connected component, and the pile is load-bearing in the exact sense the criterion means: remove it and the count is immediately two.

Cut it, and each pile end becomes two strands, each with contacts in one ground only. The bridge is gone. Two components.

The interesting thing is not that a criterion detects a cut, which would be unremarkable. It is that the fabric’s manufacture is a step from one value of the criterion to another, and that the step is deliberate, planned, and the reason the construction exists. Everywhere else on this site a fabric falling into two pieces is a fault to be caught; here it is the product coming off the loom.

Why weave it this way at all #

A velvet can be woven singly — over wires laid in as the cloth is made, which are then withdrawn or cut along a groove — and that method is older. Double plush displaced it for most purposes, for reasons that are all about the pile.

The pile length is set by the gap, not by a wire. Changing the pile height means changing the distance between the two grounds, which is a machine setting, rather than changing the wires, which is a set of parts.

Two fabrics per pass. The loom makes twice the cloth for one shed cycle, which is most of the commercial argument.

The cut is cleaner and it is the same cut everywhere. A travelling blade in a fixed plane cuts every pile end at the same height by construction, so the pile is level without shearing. A wire velvet has to be sheared afterwards to level it.

And the pile is anchored before it is cut. The pile end is bound into its ground while it is still continuous and under tension from its own beam, which is a considerably better condition to be anchored in than a free leg would be.

A criterion used three ways #

It is worth collecting the three uses in one place, because this construction completes a set and the set is the argument for having built the criterion at all.

As a filter. A draft that looks entirely reasonable describes fabric that falls into two independent layers, and nothing about the drawing betrays it. The criterion catches it. That is what the foundation essay introduced it for and it is what the four-by-four census measures: 144 drafts out of 22,874.

As a certificate. A double cloth asserted to be two must come out as two, in the right order. A test that only ever says “no” to strange things is a filter; one that will certify an intended construction and refuse to certify it as something else is a measurement. That is the second direction, and it is what makes the first trustworthy.

As a description of a process. A construction whose criterion value changes from one to two, on purpose, by a step that is part of making it. That is this essay.

The third is the one that could not have been anticipated when the criterion was written, and it is the reason a well-chosen invariant is worth more than the property it was built to check. The count was defined to answer “is this cloth?”, and it turns out to also answer “what did the machine just do?” — which is a different question that happens to have the same arithmetic behind it.

There is a fourth use lurking and it is not available. The criterion cannot say whether a cloth is about to fall apart, because there is no such state: a separation exists or it does not, and there is no margin. That is the boundary the next essays are about.

What the criterion still cannot say #

Everything the previous rungs established applies here without change, and it is worth being explicit rather than letting the elegance of the one-to-two step carry more weight than it should.

After the cut, each velvet’s pile is a set of legs anchored at one end. Whether they stay in is the anchorage question — a wrap angle, a friction coefficient, a model — and the criterion says only “attached”. A velvet with a poorly bound pile passes the criterion and sheds.

That is why the two figures on this page print the layer count in large type and nothing else. It is the one thing this construction demonstrates exactly, and it demonstrates it about the manufacture rather than about the fabric’s quality.

The relation to double cloth, which is not what it looks like #

An uncut plush looks like a stitched double cloth and is not one, and the distinction is exact.

A stitched double cloth joins its two fabrics by turning over an intersection — a face warp end passing under a back weft, say. The stitch is made of the existing thread systems, and the census of single-square changes showed that exactly half of them would join the two cloths.

A plush joins its two fabrics by a third system that belongs to neither. Remove every stitch from a stitched double cloth and two complete fabrics remain. Remove the pile from a plush and two complete fabrics also remain — but the pile was never part of either, and the fabrics are unchanged by its removal, whereas removing a stitch changes the cloth it was in.

The consequence is the one that matters industrially. A stitched double cloth cannot be separated without damaging both halves, because the stitches are made of the cloths’ own threads. A plush can, because the joining system is expendable and is meant to be cut. That is the difference between a construction designed to hold and one designed to come apart, and the criterion returns the same number for both right up until the moment it does not.

The pile density and the pile height come apart #

One practical consequence of the construction is worth drawing out, because it is the sort of thing that is obvious once stated and is not stated anywhere.

In a wire velvet, the pile height is the wire’s thickness and the pile density is how many pile ends there are and how often they are raised over a wire. Those are two settings, but they are not independent in practice: a taller wire needs more room, so the picks between wires have to be spaced accordingly, and pushing the density up at a large height runs into the wires themselves.

In a double plush, the height is the gap between the two grounds and the density is how many pile ends shuttle and how often. Neither constrains the other at all. The gap is a machine setting in the third dimension; the density is a drafting decision in the plane of each ground. A short dense pile and a tall sparse one are the same difficulty to make.

That separation is why the construction dominates. Velvet as a material is defined by a combination of height and density that a hand reads as one property — the hand of the cloth, in the trade’s own use of the word — and a construction that lets a designer move the two independently is a construction that can hit any point in that space. The wire loom can hit a curve through it.

What was counted, and how #

The construction is built as a named graph, like the single-ground pile of the first rung. Two plain grounds are generated independently and share no strand. Each pile end is given a contact beneath alternate picks of each ground and above the rest — the shuttling pattern, taken from the model rather than described in prose beside a different picture.

The cut is one line of code and it is the honest representation of a knife: the pile strand becomes two strands. Nothing else changes. Every contact that existed still exists and is attached to the same ground; the only difference is which strands they belong to.

Three assertions run, and they have to be taken together.

The uncut construction is one cloth. The cut construction is two. And a construction with the pile removed is also two — which is the assertion that makes the first one mean something, because it establishes that the pile is what was doing the joining rather than some accident of how the two grounds were generated.

The figures draw the pile from the model’s own binding pattern, so a picture showing the pile reaching both grounds and a graph saying it does not cannot both survive a build.

Where the model stops #

The gap is not modelled. The two grounds are drawn a fixed distance apart for legibility and the pile height in a real plush is that distance, halved by the cut. Nothing here computes it, because it is a machine setting rather than a property of the structure — which is exactly the situation terry’s let-off ratio is in, and the same caution applies.

Both grounds are plain and identical. They need not be, and in practice are not always: a face velvet and a backing of different construction is ordinary. Nothing in the argument depends on it.

The knife is taken as passing exactly through the middle. A blade running off-centre gives two velvets with different pile heights, which is a real and expensive fault and is the reason the cutting mechanism is the most carefully maintained part of the machine.

And the cut is taken as complete. A blade that misses a pile end leaves the two fabrics joined at that point — one cloth, by the criterion, in a place where two were intended. That failure is detectable by the criterion and it is worth noticing that this is the one construction on the site where the criterion returning one is the fault report.

What the cut costs the anchorage, exactly #

Before the knife, a pile end is one continuous strand bound into both grounds. After it, each half is bound into one. So the cut does not merely divide the yarn; it halves the number of binding points holding each piece of it.

The capstan makes that a square root. A strand wrapped through a total angle θ is held by e^(μθ); halving θ gives e^(μθ/2), which is the square root of the original. So

cutting takes the anchorage from H to √H.

At a holding ratio of 6.6 — the figure this collection computes for a well-bound tuft — the cut leaves 2.57. A factor of two and a half lost, not by any change to the binding but by the strand having become two strands.

That is the arithmetic behind an observation this collection makes elsewhere without a number: a velour sheds where a terry does not, and a cut velvet sheds where the plush it was cut from could not. The uncut construction’s pile is anchored twice over and the cut one’s is anchored once, and a square root is a large price for an operation that adds nothing and removes nothing.

It also says where the design effort has to go. A cut pile’s binding must be twice as good as an uncut one’s to reach the same anchorage, because the exponent has been halved and the only way back is to double the wrap — which is why cut-pile constructions use the deeper W-type bindings and loop piles do not have to.

And what an off-centre knife costs #

The off-centre blade is named as a real and expensive fault and left there. It has a size, and the size is uncomfortable.

A blade at a fraction x of the gap from the upper ground leaves piles of xG and (1 − x)G, so the two fabrics differ in pile height by

(1 − 2x) × G.

On a six-millimetre gap giving two three-millimetre piles, a blade three tenths of a millimetre off centre gives 3.3 and 2.7 — a twenty per cent difference in pile height between two fabrics that came off one loom and are sold as a pair.

A tenth of a millimetre of offset is six per cent of pile. So the tolerance on the blade’s position is not a machine tolerance in the ordinary sense; it is a fraction of a per cent of the gap, and it explains why the cutting mechanism is the most carefully maintained part of a plush loom.

Which gives a quality check anybody can run #

The same expression turns the fault into a measurement, because the pile’s mass goes with its length.

Weigh the two fabrics. The pile’s mass per unit area is proportional to its height, so if the pile is a fraction f of the fabric’s mass, the two pieces differ in weight by

(1 − 2x) × f.

At a pile share of sixty per cent and a twenty per cent height difference, that is a twelve per cent difference in areal mass — which is far outside anybody’s weight tolerance and is detectable on an ordinary balance in a minute.

So the blade’s offset is readable off two weighings, with no need to measure a pile height at all: x = ½[1 − (difference in weight) ÷ f], and f is known from the construction.

That is a cheap, direct check on the one setting this construction’s whole quality rests on, it uses an instrument every mill has, and it follows from nothing but the pile’s mass being proportional to its height. The two fabrics being weighed against each other is a measurement of where the knife is, which is not how a weighing is usually read.

What happens if the pile is bound to only one ground #

The construction has a failure mode that follows directly from the graph and is worth working through, because it is the case where the criterion’s verdict before the cut is genuinely diagnostic.

Suppose a pile end is drafted so that it binds into the upper ground repeatedly and reaches the lower one without ever passing beneath one of its picks — lying over it rather than under it. The pile is then attached to the upper ground and merely resting on the lower.

The criterion sees this at once. The pile has contacts in both grounds, but the directions are wrong: every edge to the lower ground runs the wrong way, so nothing in the lower ground can reach the pile, and the lower ground is a component of its own. The count is two before the knife.

That is a fault report of a very specific kind. It says: this cloth is going to come off the loom as two pieces whether or not the knife touches it, and one of them will have no pile. And it is invisible in a drawing, in the ordinary way — a pile end drawn crossing the gap looks like a pile end crossing the gap regardless of which side of the pick it passes.

So on this construction the criterion is doing all three of its jobs at once: certifying the intended two-after-cut, filtering the drafting error, and describing the manufacturing step in between. Its verdict is only informative because it is checked at both stages, which is the argument for running it before the cut as well as after.

Who found it, and when #

Face-to-face velvet weaving is nineteenth-century in its mechanised form, though the idea of weaving two cloths joined by a pile and separating them is older and appears in various hand techniques. The double-plush loom with a travelling knife is the machine that made velvet an ordinary material rather than a luxury one, and the same principle drives most modern woven pile, including a great deal of upholstery and automotive fabric.

The description in this essay — a construction defined by the value its integrity takes before and after a step — is not how anybody in the trade thinks about it, and there is no reason they should. What it buys is a way of saying precisely what the pile is for in this construction, which is not what it is for in any other: everywhere else the pile is the product and the ground holds it, and here the pile is temporary structural scaffolding that happens to become the product when it is destroyed.

Where the ladder goes next #

The pile ladder ends here. The rest of this field turns to the other construction the matrix cannot hold — leno, where the ends change places — and then states the boundary both of them sit on: the criterion is topological and cannot see friction, which is the one sentence this whole field has been assembling.