An interchange joins what a stitch would have had to

The rung below this one worked out where a stitch may be put in a double cloth and came back with an uncomfortable answer. One reversed intersection turns two cloths into one; half the intersections in the repeat would do it; almost none of them may actually be used, because a stitch that shows is a fault, and a plain-faced double cloth has nowhere at all to put one.

That whole apparatus is unnecessary the moment the design has a pattern in it.

What an interchange is #

A stitch reverses one intersection: an end that was passing under a pick of the other layer is made to pass over it, and the two plies are pinned at that point.

An interchange does something else and does not reverse anything. Across a block boundary the layer that was on the face goes to the back and the layer that was at the back comes to the face — so the same threads are still there, still interlacing with the same partners, and only which of them is uppermost has changed.

That is not an exotic construction. It is how every two-colour double cloth is designed, how a reversible coverlet gets a pattern that is its own negative on the other side, and how a matelassé, a Bolton counterpane and a two-ply blanket check are all made. The drawing a weaver uses for it is a section: two lines at two depths that cross wherever the pattern changes.

And why it joins #

Take the two-layer stacked cloth this collection uses everywhere. Every end of the face layer passes over every pick of the back layer, at every intersection between them; every end of the back layer passes under every pick of the face. Read as the above-and-below graph the criterion is built on, the arrows all point one way, and the two plies are two strongly connected components. Two cloths.

Now put a block boundary in and swap the layers across it. On the left, face-layer ends pass over back-layer picks. On the right, the same ends are now back-layer ends and pass under the picks that were the back layer’s. The arrows point the other way.

A cycle can now close. Follow an end from the left block, over a pick, and into the right block; follow that pick under an end and back into the left block. The path returns, the component is one, and the criterion says one cloth.

That is the whole of the mechanism and it is not a stitch in any sense: nothing is reversed, nothing shows, and no intersection of either face weave has been given up.

Which is measured rather than argued #

The argument above is short enough to be suspicious, so the designs are built and run through the criterion rather than reasoned about.

Five designs — two blocks across, two wide blocks across, four blocks across, a two-by-two check, and two blocks down the picks — on layers of two, three and four ends each. Fifteen drafts, fifteen times one cloth, with no stitch in any of them. And for each layer size, the same construction with no boundary at all: two cloths, every time.

The control is not decoration. A claim that a construction joins is worth exactly as much as the demonstration that its absence does not, and this collection’s own standing rule is that an assertion which has never rejected anything proves nothing.

The block boundary is a line and a stitch is a point #

The difference between the two mechanisms is worth stating in those terms, because it decides everything practical about them.

A stitch joins at one intersection. It has to be repeated across the cloth or the two plies are held at one point per repeat and slide everywhere else, and each repetition is another intersection of the face weave given up to a thread that does not belong on the face. The covering rule is about hiding those.

An interchange joins along the whole boundary. Every pick crossing the boundary carries the swap, so a block boundary a hundred picks tall joins the plies a hundred times, and none of the hundred is visible as a defect because all of them are the design.

So the two mechanisms have completely different densities. A stitched double cloth is joined at a scatter of points chosen to be invisible; an interchanged one is joined along every line the pattern draws, and the joining is as dense as the design is busy.

Which is the catch, and it is a real one #

The two facts in the last sentence go together and they are the reason the stitching apparatus exists at all.

An interchange can only join where the design has a boundary. A double cloth with no pattern in it — a plain two-ply blanket, a doubled shirting, a tube — has no boundary anywhere and no joining at all. Those are exactly the cloths that must be stitched, and they are exactly the cloths for which the hiding rule bites hardest, because a plain face has nowhere to hide a stitch.

And a design’s plain areas are unjoined. A coverlet with a large ground and a small figure is joined densely around the figure and not at all across the ground, so the ground blisters and the figure does not. That is a real fault with a real name — the trade calls it ballooning — and its remedy is to stitch the plain areas, which is to say to use both mechanisms in one cloth.

So the honest summary inverts the usual one. A double cloth is joined by its pattern where it has one and by stitches where it has not, and the stitching plan a designer draws is a map of the design’s own empty regions rather than a uniform scatter.

What the design pays for the joining #

The interchange is free in intersections and it is not free.

A block of an interchanged design is at least one repeat of the stack in each direction, because a boundary that falls inside a repeat leaves the two layers half-swapped and the construction is not the one described. On two-end layers that is a four-by-four block; on eight-end satin layers it is a sixteen-by-sixteen one. The design’s resolution is the block, and the block is set by the layers’ repeats, which is the same statement a figured cloth makes one level down.

And it costs a choice. An interchanged design has exactly two states per block — this layer up or the other one — so it is a two-tone design and cannot be anything else. Every extra tone has to come from somewhere outside the construction: a third layer, a colour order, or a shading inside each block.

A stitched double cloth has no such restriction, because the face weave is whatever the designer wants and the stitches are hidden inside it. The construction that needs no joining is the construction with the fewest design states, and that is not a coincidence: the joining is the design, so the design has to spend itself on it.

How much denser, in numbers #

“Denser” is a comparison and it is worth doing with a sett in it, because the factor is larger than the argument suggests.

Take a double cloth of two-end layers at 24 ends and 24 picks per centimetre in each layer — an ordinary cotton counterpane. A block is one repeat of the stack, four ends by four picks, so it is 1.7 mm across and there is a boundary every 1.7 mm. Each boundary joins the plies at every pick that crosses it, which is 24 per centimetre of boundary length.

Multiply: six boundaries per centimetre of width, twenty-four joins per centimetre along each, gives 144 joins per square centimetre.

Now the stitched alternative. A stitching plan that puts one stitch in each eight-by-eight repeat — which is a normal density, and denser than many — has a stitch every 3.3 mm each way, which is 9 per square centimetre.

A factor of sixteen, and it is not a close call. The interchanged cloth is not slightly better joined; it is joined so much more densely that the two constructions do not behave alike under any handling at all. That is the reason a good coverlet does not delaminate and a badly stitched backed cloth does.

The factor scales the way the argument says it should. Doubling the block size halves the boundary density and halves the joining; using eight-end satin layers rather than two-end plain ones takes the block from four ends to sixteen and the joining from 144 to 36 per square centimetre — still four times the stitched figure, on the cloth where a stitch is hardest to hide.

So the construction that needs the least joining machinery gets the most joining, and the one that needs the most gets the least. Everything about a double cloth’s practice follows from that inversion.

Two other things a boundary can be #

An interchange is one of three things that can happen where two layers meet a line, and the other two are worth naming because a draft does not distinguish them.

The layers can turn. The weft goes across the face layer, round the edge and back through the back layer, which is what happens at the selvedge of a tube and at the fold of a double-width cloth. The rung this ladder starts at is entirely about that case, and its finding was that the repeat cannot see it — a tube, a double-width cloth and two separate cloths are the same draft to the last square, and what separates them is how many free edges the finished piece has.

Or the layers can be stitched across the line. A column of reversed intersections down a block boundary joins the plies exactly as a scattered stitch does, at the cost of one face intersection per stitch. It is what a weaver does when a design has a long straight edge and the interchange would put the wrong layer on the face beyond it.

Or they can interchange, which costs nothing and forces the pattern to continue on the far side.

The three are different constructions with different costs, and the point worth carrying is that only the third is invisible in both directions. A turn shows at the edge, a stitch line shows as a line, and an interchange shows as the design — which is where it was going to show anyway.

What was counted, and how #

The drafts are assembled rather than drawn. The stacked cloth is this collection’s own stackedCloth, and its reversal is that same draft with every intersection between different layers flipped — which is what “the layers change places” means as an operation on a matrix. The design is a grid of noughts and ones and the assembly puts one or the other into each block.

The layer count comes from the same criterion as everywhere else, unchanged, and the drafts are handed to it whole rather than reasoned about block by block. That matters: a construction can be sound in each of its parts and not in the whole, which is the failure a figure that is not a stripe exists to record.

The boundaries are counted cyclically. A repeat wraps, so a design of two blocks has two boundaries and not one, and a design of one block has none. Counting them any other way would make the smallest interchange look like a single join when it is two.

The control is asserted as hard as the claim. For every layer size the no-boundary design is required to report exactly two cloths. Without it the whole result would be consistent with a stacked cloth that was one cloth all along, which — given that stackedCloth is used across this site as the standing example of a draft that is not one cloth — would have been a much larger problem than a wrong essay.

And the sweep runs over three layer sizes and five designs, because a claim demonstrated on two blocks of two-end layers is a claim about that drawing. The block size does not appear in the argument and it should therefore not appear in the result, which is a thing to check rather than to assume.

Which side is which, and why a coverlet is its own negative #

One consequence of the construction is the thing it is most often used for, and it falls out of the definition without any further work.

An interchange swaps the layers. It does not do anything else — no thread changes partners, no weave changes, and nothing is added. So the back of an interchanged cloth is the same design with the two states exchanged: wherever the face shows layer one, the back shows layer two, everywhere and exactly.

A reversible coverlet is its own negative, and it is so by construction rather than by design effort. A designer draws one side and gets the other, with the two colours swapped, at no cost and with no possibility of getting it wrong. That is a much stronger property than a two-sided cloth usually has: a damask is its own complement in the same sense, but a damask’s two sides are one cloth’s two faces and an interchanged cloth’s two sides are two different plies.

The practical difference is colour. A damask’s two tones are warp and weft of one cloth, so the two sides differ in lustre and only in lustre if the two systems are the same colour. An interchanged double cloth’s two layers can be any two colours at all, because they are separate cloths with separate warps and separate wefts — which is why the classical use of the construction is a strongly two-coloured one and the classical use of damask is not.

And there is no third option. A block shows one layer or the other, so a design has two states and its reverse has the same two swapped. Any cloth of this construction is two-coloured on both sides with the same pattern reversed, and every historical example is exactly that.

Where the model stops #

Only two layers. An interchange between three layers is a permutation rather than a swap, there are more of them, and which permutations join is not computed here. The two-layer case is the one every reversible cloth uses and it is the one this rung is about.

The section is schematic. The lower panel of each figure draws each ply as a single line, which is not what a section through a cloth looks like — it is what a weaver draws when planning an interchange. The draft above it is not schematic and carries the count; the section is there because the crossing is the thing to see and a thread-level section over four blocks is a picture nobody can read.

Nothing here is about the boundary’s appearance. A block boundary in an interchanged cloth carries a visible line — the two layers are at different depths either side of it and the transition takes a thread or two — and how visible it is depends on the yarns, the sett and the finish. The step in a figured cloth’s surface is the same class of question and its answer does not transfer, because here the two regions are the same weave at different depths rather than two weaves at one.

And ballooning is named rather than computed. That an unjoined plain region blisters is a statement about handling and washing, not about connectivity: the criterion is exact and it is blind to friction, so it cannot say how large a region has to be before the two plies separate in use. That number is the one a designer actually needs and this collection does not have it.

Who found it, and when #

The interchange is old — Bolton counterpanes, Scottish double-cloth coverlets and Jacquard matelassés all use it, and every design manual from Watson onward sets it out as a construction with a section drawing.

What the manuals do not say is that it joins. The interchange is presented as a way of getting a two-sided pattern, and the stitching is presented separately as the way of holding the layers together — two chapters, two purposes, and no statement that the first does the second’s job. A reader is left to infer that an interchanged cloth is stitched as well, which it usually is, for the reason above: the plain regions need it.

The connectivity reading is this collection’s, and it is a small argument with a clean check. What it adds to the practice is the map: stitch the regions the pattern does not visit, and nowhere else.

Where the ladder goes next #

The two layers of a double cloth have now been joined two ways and have been treated throughout as though they were made of the same warp. They are not: two layers weaving different weaves take up their warp at different rates, so they cannot come off one beam — and how much they differ is a crimp calculation that decides whether a cloth can be woven at all.