The third index is not a repeat

The compound-cloths field collected the boundary of this site’s encoding and found four ways out of it. A pile fabric declines the assumption that there are two thread systems. A leno declines the fixed order of the warp. A nonwoven declines periodicity altogether. A braid declines the right angle. Each escape loses exactly the measurements that depended on what it gave up.

It also recorded a fifth candidate and left it open:

Three-dimensional woven preforms may be a fifth escape from the encoding. They are periodic, have two systems and a third dimension, and this site has not worked out what happens to the matrix there.

It is not a fifth escape. Working out why takes one afternoon and an enumeration, and the answer is more interesting than the question: the criterion survives the third dimension entirely intact, and what the third dimension takes away is the repeat.

What an orthogonal three-dimensional weave actually is #

The construction to have in mind is the simplest of them, and its simplicity is the point.

Warp tows run along the fabric in several layers. Weft tows run across it, also in several layers, interleaved with the warps through the thickness. Neither system goes over and under the other: they simply lie above or below whichever tows they meet. An orthogonal preform contains no interlacing whatever — which is a startling sentence on a site whose founding observation is that interlacement is what makes cloth cloth.

What holds it together is a third system: a binder, or Z-yarn, running through the thickness. It comes down between two warp columns from the top surface, passes under the bottom weft, and comes back up. There are usually many of them, in a grid, and how many there are decides how firmly the preform is bound.

So the fabric is: two straight systems that hold together by nothing, and a third that supplies the whole of the connectivity. That is a strange object by this site’s standards, and it is exactly the case the criterion was built to decide.

The criterion does not care that the index is new #

The integrity criterion asks whether the threads can be split into an upper set and a lower set such that every crossing between them has the upper thread on top. Read as reachability it becomes a digraph — placing a thread in the upper set forces the thread above it there too — and the cloth hangs together exactly when that digraph is strongly connected.

Nothing in that consumes a matrix. It consumes contacts and a direction at each. The compound-cloths field found that this is why the criterion survives all four escapes: it assumes less than any of the four commitments the binary matrix makes, so one implementation decides a plain weave, a satin, a double cloth, a braid word, a warp-knit lapping, a pile fabric, a double plush and a leno gauze.

A three-dimensional preform is the fifth item on that list and the arithmetic is immediate. Levels stack, so each level’s contacts point at the level above it. A chain of nested levels has no cycle at all — from the bottom level one can reach the top, and from the top level one can reach nothing — so an unbound stack is as many cloths as it has levels. Lift the top layer off. The criterion says so, and so does anybody who has handled a dry preform.

The binder closes a cycle, and it has to reach both faces #

Add a binder and the digraph gains a node with edges in both directions: it passes under some levels, which makes it their lower thread, and over others, which makes it their upper one.

That is how the cycle closes. A binder over the top level and under the bottom one gives a path from the top down through the binder to the bottom, and the level chain carries the way back up. One cycle through every node, strongly connected, one cloth.

And it fails in two distinguishable ways. A binder that stops short of the bottom leaves the bottom ply reachable from nothing — the bottom face peels off. A binder that does not pass over the top level leaves the top ply reaching nothing — the top face peels off. Both are real manufacturing faults, both are invisible in a drawing of the preform, and both are exactly what the criterion is for.

Every binder path, counted #

Which paths work is a question with an exact answer, and the honest way to get it is to enumerate rather than to argue.

A binder either passes under a level, over it, or by it without touching — three states per level — so a stack of n levels admits 3ⁿ paths. Running every one of them through the criterion gives the count, and the count is exactly a ninth: 3ⁿ⁻² of 3ⁿ paths make one piece, at every thickness from two levels to twelve.

The rule behind it is two conditions and nothing else: under the bottom face, and over the top one. Everything in between is free, which is why the fraction is a ninth rather than something that moves with the thickness — two levels are constrained and n − 2 are not.

That is a small result and it has a practical reading. A binder that reaches both faces holds the stack however clumsily it gets there, so the design freedom in a Z-binder is almost entirely in how firmly rather than whether — which is the same division of labour the fancy weaves found between the criterion and the capstan, arriving here from a completely different construction.

The two conditions are independent, which is the useful part #

A ninth is the product of two thirds of a chance, and saying so is worth more than the fraction.

A binder path reaches under the bottom face in a third of paths and over the top face in a third, and the two events involve different levels, so they are independent. That is where the ninth comes from, and it partitions the failures exactly:

two ninths of paths fail at the bottom face alone, two ninths at the top alone, and four ninths at both.

The independence is the practical statement. A preform that is properly bound at one face carries no information about the other — knowing the binder reached the top leaves its chance of reaching the bottom exactly where it was. So an inspection that looks at one surface has learned nothing about the opposite one, and the two faces have to be checked separately rather than sampled.

That is not obvious from the section drawings, where a binder that plainly emerges at the top invites the assumption that it went all the way. The criterion says the two ends are separate questions and the arithmetic says they are uncorrelated questions.

What the enumeration cannot see is that a real preform has many binders. One short path among a grid of sound ones leaves the stack bound by its neighbours, and the criterion run on the whole structure returns one piece — correctly. The failure that matters is therefore not a bad path but a region where every binder falls short together, which is what a mis-set loom or a lifted needle bar actually produces.

So the enumeration prices a single path and the defect is spatial. That is the same shape as the pile’s tuft anchorage: the criterion answers exactly, about the object it was handed, and the object a manufacturer needs it to answer about is a population of them.

So what does the third dimension take away? #

Not the criterion. What it takes is the repeat, and this is the part worth carrying away from the field.

A weave is periodic in two directions and its matrix is a repeat: the draft tiles the plane, so every question about the cloth is a question about a small integer matrix, and the whole of this site’s arithmetic follows from that. A preform is periodic in two directions and finite in the third. It has a top and a bottom. The binder’s job is defined by those boundaries — reach the outermost levels — so the answer to “is this one piece” depends on a boundary condition rather than on a repeat.

The third index is therefore not a third dimension of the matrix. It is a stack of matrices with a first and a last, and the interesting predicates live at the ends. A fabric five levels thick and a fabric fifty levels thick have the same interior rule and different answers, because the fraction of the structure that is a face has changed.

Read against the four escapes, this is a fifth kind of departure rather than a fifth escape. The four gave up a commitment and lost the measurements that depended on it. This one keeps all four commitments in the plane and gives up periodicity in the thickness — which is the one direction the binary matrix never had.

The third system is a system this site has met twice already #

A binder is a third thread system, and a third thread system is not new here. Pile is one: a tuft standing out of a ground cloth, held by wrapping a ground pick. Terry is one, with its own beam and its own tension. A leno’s doup end is a fourth case of the same thing, and a braid’s third strand system was the first one this site drew.

The comparison worth making is with the pile, because the geometry is nearly the same and the purpose is opposite. A V-fastened tuft passes under one ground pick and comes back; a Z-binder passes under one bottom weft and comes back. Both are held by a wrap. Both are decided one-piece by the criterion. The pile’s job is to stand up out of the cloth and the binder’s is to hold the cloth down, and the arithmetic does not know the difference.

And the same silence appears for the third time. The criterion says a V tuft and a W tuft are both attached, and one of them sheds; it says a leno gauze and an open plain weave are both one cloth, and one of them comes apart in the hand; it says a preform with one binder per square centimetre and one with twenty are both one piece, and one of them delaminates. Three fields, one blind spot, and each time the response was to compute the missing quantity separately rather than to patch the criterion. A blended score would be a number with no model behind it, and that is worse than an honest silence.

Why a preform is bound at all, which the in-plane arithmetic cannot say #

There is a question the first rung of this field deliberately left, and this is where it lands. If the crimp costs stiffness and a stack of unidirectional plies has none, why weave a preform through its thickness at all?

Because a laminate’s weakness is not in the plane. It is between the plies, where there is nothing but resin: a stack of unidirectional layers loaded out of plane is a stack of layers held together by the matrix, and the matrix is a fraction of the strength of the fibre. Impact, a bolted joint, a free edge, a curved corner under load — every one of them puts a stress through the thickness, and every one of them delaminates a two-dimensional laminate.

A Z-binder puts fibre where there was resin. It is the same argument as the double plush read from the other side: there, two grounds face to face were one cloth because of the pile, and a knife made them two on purpose. Here the binder makes several layers one piece for the same reason, and nothing is meant to cut it.

So the trade is exact in its shape and unquantified in its size by anything on this site: the binder buys through-thickness strength and spends in-plane stiffness, because every millimetre of binder is fibre not running along the load, and because the tows it passes between are locally crimped by it.

What was counted, and how #

The enumeration is over every path in {under, over, past} per level, which is 3ⁿ, run through the same stronglyConnected that decides every other structure on this site. At seven levels that is 2,187 paths, each producing a digraph of eight nodes.

Four things are asserted rather than described. The count of one-piece paths equals 3ⁿ⁻², at every thickness tested. An unbound stack has exactly n pieces. A through-thickness binder gives one piece. And a binder that stops half way does not — which is the assertion that would catch a digraph built with its edges the wrong way round, because a reversed edge would make the half-depth binder work.

One detail of the model is a decision rather than a fact, and it is stated in the code. The binder is a node only when it touches something: a binder with no contacts is not a loose thread the criterion should count, it is a thread that is not in the fabric, and counting it would report one piece too many for the empty path. That is the kind of choice that silently shifts a census by one, so it is written down beside the enumeration.

What the picture cannot show #

The preform figures draw a section and a digraph, and neither can show the quantity a fabricator would ask about first.

They cannot show how many binders there are. The section draws one column and the criterion consumes one node; a preform with a binder every second column and one with a binder every eighth are the same picture and the same verdict, and one of them delaminates. That is the criterion’s silence again, and it is the same silence the pile and the milling fields met.

They cannot show the in-plane cost either. A Z-binder crimps the tows it passes between, and crimp costs stiffness — but the local crimp a binder imposes is a three-dimensional deformation that a section through one column does not carry.

What the pair of drawings does carry is the identity of the object: a section and a digraph, the same contacts read two ways, so that the criterion deciding a preform can be seen to be the criterion that decides a plain weave.

Where the model stops #

One node per level is a simplification with a consequence. The parallel tows within a level have identical contacts in this model, so the criterion cannot see a binder that holds only some of the columns — and a real preform with binders every fourth column has a partially bound structure whose separability depends on how the levels are tied laterally, which they are not. The model answers the question about a column and calls it the fabric.

The criterion still cannot see firmness. A preform bound by one binder per square centimetre and one bound by twenty return the same answer, and one of them delaminates. That is the third time this site’s central instrument has been silent in a way that turned out to be the finding, and the response is the same as it was for pile and for felting: compute the missing quantity separately and print the two side by side, rather than extending the criterion into a number with no model behind it.

And nothing here computes what a Z-binder costs. It displaces the in-plane tows it passes between, it crimps them locally, and it is itself off-axis reinforcement doing very little for the in-plane stiffness. Every one of those is a knock-down of the kind the first rung of this field computed, and none of them is computed here.

Who found it, and when #

Three-dimensional weaving as a manufacturing technique is from the 1970s and 1980s, and it arrived from the aerospace side rather than from weaving: the demand was for a near-net-shape preform that would not delaminate, and the looms were adapted to supply it. The classifications came afterwards — orthogonal, angle-interlock, layer-to-layer, through-thickness — and they are classifications of where the binder goes, which is exactly the parameter the enumeration above is over.

What is worth noticing is that the trade’s taxonomy and the criterion’s answer are about different things. The taxonomy distinguishes constructions by the binder’s path in detail, because the path decides the stiffness, the crimp it imposes on the tows it passes, and how hard the preform is to weave. The criterion collapses all of that to two bits: does the path reach the bottom face, and does it reach the top. A classification with a dozen names and a predicate with two conditions, over the same object — and neither is a substitute for the other, which is the ordinary relationship between a topological fact and an engineering one.

Where the ladder goes next #

The applied field turns now from cloth built to be filled to cloth built to let something through on purpose. The quantity is one this site has computed since it first asked how close threads can be set — the clear gap between two of them — and the trade that lives on it specifies a fabric by two numbers that pull opposite ways. The next rung draws the hole between four threads and measures it against the thing it has to hold back.