A tube and two cloths are the same draft

A narrow loom can weave a wide cloth. It can also weave a bag with no seam in it, and it can weave two entirely separate fabrics at once and drop them off the front beam in a heap. All three are done with two layers of cloth in the same repeat, and all three come off the same loom with the same shafts lifted in the same order.

The question this essay is about is how to tell which of the three is being made. It is not answerable from the draft, and the reason it is not answerable is worth more than the answer.

Three cloths that are one draft #

A double cloth is two complete fabrics woven at the same time. Alternate ends belong to the face and the back, alternate picks likewise, each system interlaces with its own opposite number, and the two pass each other without ever exchanging: face warp always over back weft, back warp always under face weft.

That construction has three ordinary uses and they are not variations on each other. They are different objects.

Two separate cloths. The layers are never joined at all. What comes off the loom is two fabrics, each of the loom’s own width, lying on one another and connected nowhere. It is how a mill weaves two light linings in the time it takes to weave one.

A double width. The layers are joined at one selvedge. The cloth comes off folded, is opened out, and is twice the width of the loom — which is the trick every hand loom narrower than a blanket has depended on since long before anybody wrote a matrix down.

A tube. The layers are joined at both selvedges. The cross-section closes, and what comes off is a seamless cylinder: a fire hose, a sack, a bolster cover, the woven part of a stent.

What the criterion is a statement about #

The integrity criterion decides whether a draft describes one cloth or several, exactly and in one linear-time pass. Draw a directed edge from the lower thread to the upper one at every crossing; the fabric hangs together exactly when that digraph is strongly connected, and the number of separable cloths is the number of components.

Run it on the draft above and it returns two. Run it on the tube and it returns two. Run it on the double width and it returns two. All three answers are correct.

The reason is that the criterion is a statement about a repeat tiled over the whole plane, and a plane tiled with a repeat has no edges anywhere in it. The word “selvedge” cannot be expressed in the object the criterion is about. This is the same boundary the criterion cannot see friction records from the other side: the check is exact, and exactness is a property of the question it is asked, not a guarantee that the question is the one somebody wanted answered.

It is also, in miniature, what a repeat repeats is about. A draft carries the cloth and a decision about how much paper to use; here it carries the cloth and omits a decision about what happens where the paper stops.

The finite fabric, and the two numbers it has #

The repair is not a better criterion. It is a different object: a finite fabric, of a stated width, with a selvedge at each side and a rule at each selvedge saying what the weft does on reaching it.

There are two such rules and no others. The thread turns — comes back in the layer it was already in — or it crosses, and comes back in the other one. A turn leaves that layer with an edge of its own. A cross makes the two layers share an edge, and a shared edge is not an edge.

The consequence for the connectivity is immediate and it is the whole mechanism of this essay. In a repeat, every pick is an independent strand: a face pick and a back pick are different threads and can never be the same thread. In a piece of cloth they are the same thread whenever the weft crosses at the selvedge between them — and merging two nodes of a digraph is not a small change, because it merges their components.

So a finite two-layer fabric has two numbers that its repeat does not have.

The first is the number of pieces: the components of the same above-and-below digraph, computed with each continuous weft thread as one node rather than each pick. The second is the number of free edges: how many selvedge lines the finished cloth has that are not folded into another layer, which is two for every selvedge at which the weft turns and none for every selvedge at which it crosses.

The four cases, and what separates them #

Neither number separates the constructions on its own, and that is worth being exact about rather than glossing.

The piece count separates two loose cloths, which is two pieces, from everything else, which is one. It cannot tell a tube from a double width, and it cannot tell either of them from an ordinary double cloth stitched together at one intersection per repeat — all three are a single piece of cloth and correctly so.

The free-edge count separates the tube, at none, from the double width, at two, from both of the flat cases, at four. It cannot tell two loose cloths from a stitched double cloth, because stitching happens in the middle of the cloth and changes no edge.

The pair separates all four exactly, and the enumeration below is the check that it does rather than the claim that it should.

It is worth being clear about what kind of statement that is. Two numbers separating four things is not a theorem about all two-layer fabrics; it is a measurement over the four constructions anybody actually weaves, and a fifth construction could perfectly well collide with one of them. What the pair does guarantee is that the two quantities are independent — the piece count and the free-edge count are not two readings of one thing, because each of them takes two values while the other is held fixed. A single number could not do it, and the reason the site had no number at all until now is that both of the available ones live in an object the repeat is not.

What was counted, and how #

Every number in this essay is the output of building the finite fabric and measuring it while the figures are drawn.

The join rules are turned into a shuttle path first, because the pick order is not free. A pick is laid in one layer travelling one way, so the shuttle’s state is a layer and a direction — four states in all. Reaching an edge reverses the direction always and the layer only if that edge folds. That is a permutation of four states, and its orbits are the threads.

Reading the orbits off gives three results that were not put in.

A tube needs one shuttle and two separate cloths need two. The tube’s orbit visits both layers, so one thread weaves the whole fabric by spiralling; with both selvedges turning, neither orbit leaves its own layer, so a second shuttle is unavoidable. The cheapest of the three constructions at the loom is the one that sounds hardest.

A tube’s counter-spiral is never thrown. Two of the four states are idle in a tube, being the spiral that runs the other way round. Nothing is wrong with it; it is simply a second tube that nobody is weaving.

Two of the three constructions are literally the same matrix. The tube and the two loose cloths both lay picks face, back, face, back, and their drafts agree in every square. The double width does not: its single shuttle has to reverse, reversing costs a doubled pick in one layer, and its pick order is face, back, back, face.

The connectivity is then computed on the finite object: ends as they are, and one node per weft thread rather than one per pick. Four constructions, four distinct pairs of numbers, and the assertion that they are four distinct pairs is made while the table is drawn rather than after it. The assertions run the other way too — a construction claiming to be a tube while a selvedge still turns is refused, which is the mistake this whole apparatus exists to make impossible.

The width is the same in all three #

Here is the result with no free parameter in it, and it is an arithmetic triviality that stops being trivial the moment it is said out loud.

A loom of width w weaving two layers produces, per pair of picks, exactly 2w of cloth. That is true whatever the selvedges do, because the weft crosses each layer once and there are two layers.

So a tube off that loom has a circumference of exactly 2w. A double width opens to exactly 2w. Two separate cloths are two pieces of w, and 2w between them. The developed width is the same in all three cases and the topology is not.

A 90 cm loom therefore gives a 180 cm blanket, or a tube 180 cm round — which is about 57 cm across if it is laid out circular — or two 90 cm linings. The cloth is the same cloth and the same weight of yarn goes into it. Only the edges differ, and the edges are free.

That is the practical form of the whole argument. A weaver choosing between the three is not choosing a structure, a sett or a yarn. They are choosing what the weft does twice per pick pair, at two places, and everything else about the cloth follows from a decision that takes up no room in the draft at all.

More than two layers, and the condition for a single tube #

The shuttle-path argument is written for two layers and it does not depend on there being two. Generalising it says how wide a tube a narrow loom can weave, and it turns up a parity condition nobody would guess.

With k layers the shuttle’s state is a layer and a direction — 2k states. Reaching an edge always reverses the direction, and moves the shuttle between layers according to that edge’s own rule, which is a pairing of the layers: some pairs fold into one another and any layer left over turns.

So each selvedge contributes a matching on the k layers, and the fabric’s connectivity is decided by the union of the two matchings read as a graph. If that graph is connected, the cloth is one piece and one shuttle weaves the whole of it; if it falls into components, each component is a separate cloth needing its own shuttle.

Three consequences, and the last is the parity one.

A tube of k layers has circumference kw. The width argument does not care how many layers there are: a loom of width w laying k picks produces kw of cloth, and if the two matchings connect every layer into one path the whole of it closes into a single cylinder. A 90 cm loom weaving six layers gives a tube 5.4 m round — which is a bag no loom of that width could otherwise make.

The layer count is capped by the repeat. A repeat of n ends holds at most ⌊n/2⌋ separable cloths, so a six-layer tube needs twelve ends in its repeat and twelve shafts to weave it. The circumference a narrow loom can reach is therefore its width times half its shafts — which is a statement joining a harness count to a finished dimension with nothing in between.

And a tube of an odd number of layers is impossible. A matching pairs layers two at a time, so a perfect matching needs k even; with k odd, at least one layer is unpaired at each selvedge and turns, and a turn leaves a free edge. So three, five and seven layers cannot close, whatever the join rules, and the tubes a loom can weave have two, four or six layers and no others.

That last is a genuine restriction and it has the right shape for this collection: it is a parity argument, it involves no yarn and no machine, and it explains a piece of practice that is otherwise a convention. Multi-layer tubular constructions in the trade — hose, sleeving, layered preform sacks — come in even layer counts, and the reason is that an odd one is a tube with a slit down it.

The caution is the same as the two-layer case’s, and it grows with k. The matchings say which layers meet at an edge and nothing about whether the fold is fair, and a six-layer selvedge is folding three pairs of cloth into one edge. That is a great deal of yarn in one place, and whether it makes an edge anybody would accept is a question about the weft’s turn and not about the graph.

Where the model stops #

A join rule is a statement about which layer, not about how. Whether the weft turns tightly enough to make a fair fold, whether the fold is thicker than the cloth either side of it, whether a tubular selvedge draws in — none of that is here, and all of it is what a weaver actually worries about. The fold in the figure is a line with no width.

The layers are given no separation. The section drawing puts a gap between them for legibility, as every crimp figure here does, and the matrix has no notion of thickness at all. A spacer fabric and a limp double cloth are the same object to everything on this page.

The finite fabric is finite in one direction only. It has two selvedges and no cut ends, which is right for cloth on a beam and wrong for a made-up article. Adding the ends would change the piece count for nothing and the free-edge count in a way that is about cutting rather than about weaving.

And the piece count is coarse. One piece of cloth is one piece of cloth whether it is held together by a fold at the selvedge or by a stitch in the middle, and the mechanical difference between those two is enormous. That is the criterion’s standing limit again, in the one place where a second number happens to be available.

Who found it, and when #

None of the constructions is new and none of them was ever confused in practice. Tubular weaving is at least as old as woven bags, double-width weaving is what every narrow hand loom did when a blanket was wanted, and the two are described correctly, with drawings, in every weaving manual there is.

What the manuals give is a picture of the shuttle path, which is exactly right and exactly the thing a matrix has no room for. The trade’s own notation solved the problem by not using a matrix: a tubular draft is given as a treadling plus a sentence, and the sentence is where the tube is.

The formal side is more recent. The connectivity criterion belongs to the work on periodic fabrics of the 1980s — Grünbaum and Shephard’s — and it is a theory of periodic fabrics by construction, so the selvedge falls outside it by design rather than by oversight. Reading that omission as a gap to be filled with a finite model is this site’s move and not theirs.

Two of this site’s own essays state the joins the other way round, giving the tube one selvedge and the double width both. That is the correction this essay makes, and it is exactly the kind of error a site that verifies its drawings and not its sentences will make: the drafts were right, the layer counts were right, and the two names were swapped in a clause that nothing checked.

Where the ladder goes next #

The next rung takes up the question this one raises and leaves alone: a stitch joins the layers, one intersection per repeat is enough, and where it goes is decided by whether the face weave can cover it. That turns out to have an exact answer, a rule that has to be written down before it can be counted, and a face weave with nowhere at all to put one.

Sideways from here, the same two-layer construction is what a backed cloth and a reversible coverlet are made of, and the layer count is a design specification rather than a check in both. Further out, the stack goes deeper: a three-dimensional preform is the same rule at every crossing between systems with more systems to apply it to, and how many layers a repeat of a given size can hold turns out to have a hard ceiling.