Backed and stitched constructions

Everything so far has treated a cloth as a plane: two thread systems, one crossing the other, one layer. A great many useful fabrics are not like that. They are two complete cloths woven at the same time, on the same loom, in the same repeat — and the question of how the two are related turns out to be the only interesting thing about them.

This is the direction of the integrity check that earns it its trust. A test that only ever says “no” to strange things is a filter. A test that certifies an intended two-layer construction as two, and refuses to certify it as one, is a measurement — and it is the only kind that can catch the mistake that matters here.

How a double cloth is put together #

The construction is easier than the name suggests. Take alternate warp ends for the face and alternate ends for the back; take alternate picks likewise. Then four kinds of intersection exist, and each has a rule.

Face warp meeting face weft: interlace them normally, in whatever weave the face is to be.

Back warp meeting back weft: the same, in whatever weave the back is to be. The two need not match; a satin face on a plain back is an ordinary thing to want.

Face warp meeting back weft: the face warp always passes over. It has to, or the face warp would show on the back.

Back warp meeting face weft: the back warp always passes under, for the same reason in reverse.

Those last two rules are what make it two cloths. Every cross edge in the above-and-below digraph runs the same way — from the back system towards the face system — so the face is reachable from the back and the back is not reachable from the face. That is precisely a two-component digraph, and lifting the face away is exactly what the mathematics says is possible.

It is also exactly what happens if the cloth comes off the loom unstitched. Two fabrics, in the same space, connected nowhere.

One stitch #

A double cloth is only useful if something joins it, and the joining is a small change.

Turn one intersection over — let one back warp end come to the face over one face pick, or let one face end drop under one back pick — and a cross edge now runs the other way. The digraph becomes strongly connected. The cloth becomes one cloth.

One. Not a row of them, not a scattering — one intersection per repeat is sufficient, and the arithmetic is not subtle about why: the digraph needed exactly one edge in the missing direction.

That is the forward reading, and it explains the whole family of constructions. A backed cloth — an ordinary fabric with an extra warp or weft carried on the reverse for weight or warmth — is a double cloth stitched sparsely enough that the back never shows through. A quilted or matelassé cloth is stitched at intervals wide enough to leave the layers free between them, which is what gives the puckered surface. A tube is a double cloth left unstitched across the width and joined at both selvedges; a double-width cloth is the same thing joined at one, opened out after weaving to give a fabric twice as wide as the loom.

Every one of those is the same repeat with a different stitching pattern, and every one is verified by asserting a layer count rather than by looking.

The enumeration, and the direction that matters #

Now the backward reading, which is the reason this belongs on the integrity ladder rather than in a catalogue of constructions.

If one intersection joins two cloths into one, then one mistake destroys a double cloth. And the question of how many of the sixteen intersections in the smallest repeat would do it can simply be answered, by trying each of them in turn and running the check.

Exactly eight of sixteen. Half. And at the next size up, eighteen of thirty-six — half again.

The number has a clean reason once seen. The intersections split into two kinds: those where a face thread meets a face thread or a back meets a back, and those where the two systems cross. Reversing one of the first kind changes the face or the back weave and leaves the cross structure alone, so the cloth is still two. Reversing one of the second kind flips a cross edge, and every cross edge is one of the eight the construction relies on. Half the squares are load-bearing and half are not, and nothing about the drawing distinguishes them.

For a designer this is a sharp result. A jacquard double cloth may have a repeat of several hundred ends and several thousand intersections, half of them positions where a single wrong cell silently converts a reversible two-colour coverlet into a heavy single cloth — or, in the other direction, converts an intended stitching into a place where the layers separate.

Backed cloth, and why the back is different #

A backed cloth deserves separating from a true double cloth, because the two are made the same way and used for opposite reasons.

In a double cloth both faces are faces. The construction exists to produce two surfaces, usually different, and often to swap them according to a pattern so that a two-colour design appears in reverse on the back. Coverlet weaving is built on this.

In a backed cloth the back is a passenger. It exists to add weight, warmth, or a soft handle against the skin, and the design requirement is that it should not show through. That constrains the stitching in a way the double cloth’s does not: the stitch points must be placed where the face weave hides them, which in practice means under a face float and never at a face interlacing.

That is a real design constraint with an arithmetic form: the stitching pattern must itself be a scattered set, for exactly the reason a satin’s interlacings are scattered. Stitch on a regular grid aligned with anything and the stitches read as a pattern of their own; stitch on a satin-like distribution and they disappear.

Where the stitch may go #

The enumeration says half the intersections would join the cloth. It does not say a designer may use any of them, and the difference between “would work” and “may be used” is where the craft lives.

A stitch is a thread appearing where it does not belong. A back warp end brought to the face shows a spot of the back colour on the face; a face end dropped to the back shows a spot of the face colour on the back. Both are visible, and the whole art of stitching a double cloth is making them not be.

Three devices do it, and each is a constraint on which of the eight positions is chosen.

Hide it under a float. A stitch placed where the neighbouring face ends are on the surface is covered by them. That is why a satin or twill face is easier to stitch invisibly than a plain one: plain weave has no float anywhere to hide under, which is why plain-faced backed cloths are rare.

Scatter them. Stitch points on a regular grid produce a pattern of their own — a faint check the designer did not ask for. Distributing them on a satin-like arrangement, so that no two fall on the same end or the same pick within the repeat, makes them invisible for exactly the reason a satin’s interlacings are invisible.

Use the direction that shows least. Stitching from the back upward puts back-coloured spots on the face; stitching from the face downward puts face-coloured spots on the back. Which matters depends on which side is seen, and a cloth with two seen sides has to compromise.

None of those three is in the arithmetic. The arithmetic says eight positions join the cloth; the craft says perhaps two of them are usable in a given design, and which two depends on the face weave. That is the ordinary relationship on this site between a decidable property and a design decision: the computation narrows the field and does not choose within it.

What the layers do that one layer cannot #

Two properties come out of the construction and neither is available to a single cloth.

Thickness without coarseness. A heavy single cloth needs heavy yarn, which makes it thick and coarse. Two light cloths stitched together give the same weight in more thickness, and layers free to slide over one another bend much more readily than a solid of the same thickness — the principle that makes a stack of paper more flexible than a board. Whether they bend more readily than the single cloth of the same weight is a different question, and the answer depends on the yarn: if its fibres slide, dividing the same yarn into layers leaves the bending stiffness exactly where it was, and only a set yarn makes the double cloth the softer of the two.

Two independent surfaces. The face and the back can carry different weaves, different colours and different fibres. A wool face with a cotton back is a normal construction; so is a cloth whose face is warp-faced satin and whose back is weft-faced, so that the two sides wear differently.

The second property is where the layer count becomes a design tool rather than a check. Exchanging the layers according to a pattern — bringing the back to the face in some regions and not others — gives a reversible two-colour cloth, and every region of exchange is a place where the connectivity has to be got right.

Three layers, and the fabric that is mostly air #

The construction generalises upwards without any new idea, and the generalisation is where it stops being a clothing technique.

A triple cloth is three systems rather than two, with the same rule at every crossing between systems: the upper always over the lower. The digraph has three components and the check asserts three. Stitching then has to join three things rather than two, which needs at least two reversed intersections and not one — the same argument, one component further along.

Past three, the interesting case is the one where the layers are deliberately held apart. A spacer fabric is two cloths joined by pile threads standing between them, so the fabric has a thickness that is mostly air and a compressive stiffness that comes from those threads buckling rather than from any material. Mattress ticking, sports shoe uppers and much protective padding are made this way.

Topologically a spacer fabric is a stitched double cloth, and the check reports one. What the check has nothing to say about is the thing that makes it useful, which is the distance between the layers — a quantity fixed by the length of the connecting threads and entirely absent from the matrix. It is a good example of the site’s standing caution: a construction can be completely characterised by the criterion and still have its whole point lie outside it.

Woven three-dimensional preforms for composites are the same family taken further: many layers, connected by threads passing through the thickness in a designed pattern, so that the finished part cannot delaminate the way a stack of two-dimensional plies can. There the layer count is not a check on a construction but the specification itself, and what happens when such a cloth is laid over a mould is a separate problem with its own limits.

Where the construction came from #

Double cloth is not a modern refinement. It is old enough that its several independent inventions can be identified.

The tubular and double-width uses are the practical ones and they answer a constraint of the loom: a narrow loom can make a wide cloth if the cloth is folded, and the fold is woven rather than sewn. That trick appears wherever hand looms are narrow, which is nearly everywhere before the nineteenth century.

The reversible two-colour use is the decorative one, and its most developed form is the American and British coverlet tradition of the eighteenth and nineteenth centuries. A coverlet is a double cloth in two colours whose layers exchange according to a pattern, so that the design appears dark-on-light on one side and light-on-dark on the other. Every exchange is a place where the connectivity must be right, and the drafts were worked out by hand.

The jacquard made the exchange arbitrary. Once every warp end can be lifted independently, the pattern of exchanges is a picture rather than a threading, and the constraint moves from what the loom can do to what the structure permits — which is where the arithmetic in this essay starts to matter, because a designer drawing a picture has no mechanical feedback about whether the cloth still holds together.

The formal treatment is much later and belongs to the same 1980s work on periodic fabrics that produced the criterion itself. What that work supplied was not the construction, which was thoroughly known, but the observation that “how many cloths is this” is a well-posed question with a computable answer.

How close the stitches have to be #

The first limit — that one stitch per repeat makes a cloth one cloth and does not stop the layers sliding between stitches — is stated as a mechanical question with no answer in the matrix. It has an answer, and the answer is a length.

When a two-layer sheet bends to a radius R, the two layers’ neutral surfaces are a layer thickness t apart, so they must slide past one another at a rate t/R per unit length. Between two stitches a distance s apart, the slip accumulated at the midpoint is

slip = (s ÷ 2) × (t ÷ R).

The layers blister when that slip exceeds what the structure can absorb, which is of the order of a yarn diameter d. Setting the two equal gives the stitch interval:

s = 2Rd ÷ t.

Put ordinary numbers through it. A cloth of 0.3-millimetre layers stitched with a 0.2-millimetre yarn, bent to a five-millimetre radius — a sharp fold at a lapel or an elbow — needs stitches every 6.7 millimetres. The same cloth bent to a fifty-millimetre radius, which is ordinary drape, needs them only every 67.

Both of those are the intervals the trade uses. A backed suiting is stitched every few ends — millimetres — because it is expected to be folded; a quilted or matelassé cloth is stitched at twenty to eighty millimetres because it is expected to hang. The two practices look like different traditions and they are one expression evaluated at two radii.

The scaling is the useful part. The stitch interval goes as the bend radius over the layer thickness, so a heavier double cloth needs proportionally closer stitching for the same service — which is why the heavy reversible coverlets are stitched so much more densely than their patterns require.

And the fraction that is load-bearing grows with the layers #

The enumeration finds half of the intersections load-bearing at two layers and half again at the larger repeat, and the essay’s reason generalises to a formula.

With n systems, each taking one end in n and one pick in n, the intersections where a system meets itself are n × (1/n)² = 1/n of the total. Every other intersection is a crossing between systems, so

the fraction of load-bearing intersections is 1 − 1/n.

Two layers: a half, which is the census’s number. Three layers: two thirds. Four: three quarters. And in the limit of many layers, essentially all of them.

So a multi-layer construction is more dangerous per square, not less. A triple cloth has two thirds of its draft in positions where one wrong cell changes the layer count, against a double cloth’s half — and a woven three-dimensional preform, which may have six or eight layers, is approaching the state where almost every intersection is load-bearing.

That is a sharp warning for exactly the constructions where the layer count is the specification rather than a check. The more layers a designer asks for, the larger the fraction of the design that is silently structural, and the fraction rises towards one while the number of stitches needed to join them rises only as n − 1.

Two things moving in opposite directions: more to get wrong, and fewer places where getting it right is enough.

What the check does not know #

Three limits, and they matter more here than usual because a double cloth is a mechanical object in a way a single cloth is not.

It does not know about slippage. One stitch per repeat makes the cloth one cloth topologically. It does not stop the two layers sliding relative to one another between stitch points, which is a real behaviour and the reason quilted cloths bag and blister. Whether a stitching interval is close enough is a mechanical question with no answer in the matrix.

It does not know which side is the face. The construction is symmetrical; the designation of face and back is a decision made outside the draft, and reversing it gives a perfectly valid and completely different cloth.

It does not know about the edges. Everything here is stated for the infinite periodic cloth. Whether a tube is a tube depends entirely on what happens at the selvedges, which is where the two layers either join or do not — and the selvedge is not in the repeat.

Where the ladder goes next #

This is the top of the integrity ladder as it stands. Below it are the nonwoven, where the criterion has nothing to work on, and the braid, where it applies to a single oblique thread system, and at the base the woven case it was built for.

The construction’s own companions lie in other fields. The bending advantage of layers is a mechanics question; the balance of the two faces is a setting question; and the pattern possibilities of exchanging the layers belong with colour and weave, where the surface again fails to report the structure beneath it.