A figure is not a stripe

An earlier essay here proved something reassuring. Take two weaves, each of which is a perfectly good single cloth, and thread them in bands across the warp — a satin stripe on a plain ground, a corded stripe, a shirting with a hairline. The result is always one cloth. Not usually, not almost always: always, unless a band was cut narrower than its own repeat, and even then a neighbour often rescues it.

The proof is two lines. Each band’s ends, together with all of the picks, form that band’s own above-and-below digraph, which is strongly connected because the band is a sound weave. Every band meets every pick. Two strongly connected subgraphs sharing a vertex are strongly connected. So the whole is.

That proof has a hole in it the size of a damask.

The move a figure makes that a stripe does not #

A stripe partitions the ends. A figure partitions the intersections.

That is the whole difference and it is not a technicality. Damask, huckaback, summer and winter, M’s and O’s, and every jacquard design that is not a stripe put one weave inside a region of the point paper and a second weave outside it, and the boundary of that region runs in both directions at once. A block of cloth in the middle of a figure meets neither all the ends nor all the picks. The stripe proof has nothing left to stand on.

So the question has to be asked again from the beginning, and asking it by enumeration rather than by argument is what this essay is.

The object being enumerated is the weaver’s own. A block design is drawn first as a profile — a small grid in which one square stands for a whole block of cloth — and the weaves are filled in afterwards. Every figure on this page carries its profile above it at one square per block, so the small grid and the large one are visibly two scales of one object.

What the block rule says, and why it is a theorem #

The first result is the reassuring one, and it is worth having because it bounds the damage.

A figure is coarse for a pair of weaves when its block is a whole number of repeats of each, in both directions. Under that condition the stripe proof can be run twice.

Inside one column of the profile, each block is a complete piece of its own weave over that column’s ends and its own row’s picks — so it is strongly connected on its own. Blocks stacked in one column of the profile share all of that column’s ends, so their union is strongly connected. And columns of the profile share all of the picks, so the whole is. A figure of sound weaves at a coarse block is a stripe of stripes, and it cannot separate.

Below that resolution it can. The next figure is the same edge drawn at three block sizes on a pair whose repeat is eight, and what changes is not how the edge looks.

The three ways a figure fails, and only one of them is invisible #

Running the census over five pairs of weaves and all 65,536 profiles apiece — 327,680 figured cloths in all — sorts the failures into three kinds that behave completely differently.

Most failures are visible. 48,066 of them leave a thread lying loose: an end or a pick that the boundary happens to keep on the face for the whole repeat, which is a float a weaver would reject before anybody computed anything. These are failures of drawing rather than of structure, and they are the reason the trade’s rule about stepping a figure boundary exists at all.

A few are stripes wearing a figure’s clothes. Twenty-eight of the profiles that separate are the degenerate ones whose blocks happen to line up in whole rows or whole columns — a band of half a repeat, which an earlier essay here already settled.

And 1,545 are neither. A genuine two-dimensional figure, no thread loose, no long float, both weaves sound, and the cloth in two pieces.

That last point is the one worth sitting with. When a draft describes two cloths, the intuition is a stack — a face fabric and a back fabric, one above the other, which a cross-section would reveal. Here the two cloths are interdigitated. Every other end belongs to the upper set and every other pick with it; pull, and what comes away is a fabric with the same construction as what stays, occupying the same space, and half the thread count.

The number that decides it is not in either draft #

Here is the finding this essay was written for, and it was not the one the enumeration was set up to look for.

The pair above is a 2/2 twill figured on a 3/1 twill. Both are ordinary. Now take the same two weaves and slide the ground one end along before blending. Every measure this site takes off a matrix is unchanged: same floats, same interlacings, same balance, same shafts, same lifts, same layer count. It is the same cloth; only the writing moved, and which rectangle a draft is written on is a decision by whoever drew it.

Sweep all sixteen relative origins and run the whole 65,536-profile census at each. Eight of the sixteen are safe at every figure there is. The other eight break on more than a third of them.

The pattern in the sixteen is exact and it collapses to one number. A 2/2 twill is unchanged by sliding one end and one pick together, because that is what a step-one twill is. So only the difference between the two offsets matters, it takes four values, and two of those four are safe.

Which is to say: the quantity that decides whether two weaves may be figured together is a single integer, mod four, that no drawing of either weave contains. Point paper has no origin. A draft is written on a rectangle whose top-left corner is wherever the pencil went down, and this site has a whole essay on why that rectangle is a decision rather than a fact. All of that is true of a weave on its own. The moment two weaves meet in one cloth, their relative origin is a real quantity — the only part of the pair of origins that cannot be slid away — and it decides the answer.

A designer choosing a ground weave and a figure weave is making a decision they know they are making. A designer deciding where on the point paper to start writing the ground is not making a decision at all, as far as they know.

What was counted, and how #

Everything above is an exhaustive enumeration; nothing is sampled.

A profile is a k by k grid of blocks, so there are 2^(k²) of them, and every census here is run at k = 4, which is 65,536. For each profile the composite matrix is built by tiling both weaves from a common origin and choosing between them cell by cell, and then the connectivity is computed on the result. The panel of five pairs is 327,680 composites; the phase sweep is sixteen full censuses, one per relative origin, and is 1,048,576.

Two things about the tiling are load bearing and are worth stating rather than assuming. The two weaves are tiled from the same origin, because a loom does not restart a weave at a boundary — it lifts whatever the design says on the pick it has reached, so an end’s phase in the ground is decided by where it is in the cloth and not by where the figure ends. And the repeat used in the block rule is each weave’s own minimal repeat, computed rather than read off the writing, because a draft written larger than it repeats would make the rule look stronger than it is.

The inner loop is written twice. The connectivity computation builds a validated matrix and runs Tarjan over it, which at a million composites is minutes; the fast path represents the digraph as a bit mask per node, assembles it from a per-block table, and decides strong connectivity by forward and backward reachability from one vertex. Only the profiles the fast path rejects are given the full pass. The two are required to agree on every profile of a smaller grid, in both directions — which matters, because a census only ever asks the slow path about profiles the fast one has already rejected, so a fast path that wrongly called something sound would undercount in silence and stay green.

Three assertions guard the results rather than decorate them. A coarse block that ever produced a separation would falsify the block rule, and the census refuses to return if one does. A sweep over the origins that came back flat would mean the relative origin had been divided out somewhere in the construction, and the sweep refuses to return unless it finds both safe and unsafe origins. And the panel refuses unless the visible failures outnumber the invisible ones, because the interesting claim is that the invisible case is rare and an assertion that cannot fail proves nothing.

How many relative origins a pair actually has #

The sweep above ran sixteen origins on a pair of four-end twills and found four distinct answers. That collapse was explained by hand — a step-one twill is unchanged by sliding one end and one pick together — and it generalises into a count that can be made before any census is run, which matters because a census is expensive and a count is not.

An origin is a pair of offsets, so a pair of weaves on repeats r by r has r² of them. Sliding one weave by a translation that maps it to itself changes nothing, and neither does sliding the other one that way, so what actually varies is the offset modulo the translations both weaves possess. Write G for the group of translations under which a weave is unchanged. Then

the number of genuinely distinct pairings is r² divided by the order of the intersection of the two weaves’ translation groups.

Check it against the case in hand. A step-one twill on four ends is unchanged by the translation (one end, one pick), and that translation has order four, so G has order four for both weaves; the intersection is the whole of it; and 16 ÷ 4 is four. That is the number the census found, arrived at without running it.

Now a pair the census also covered. An eight-end satin with a move of three is unchanged by (one end, three picks), which has order eight. Its reverse — the sateen used as the ground — is the same weave turned over and has the same translation group. So 64 ÷ 8 is eight distinct relative origins for a satin-on-sateen figure, not sixty-four.

And the extreme case, which explains why nobody has ever noticed the effect on the commonest figured cloths. A plain weave is unchanged by (one end, one pick) and by (one end, minus one pick), so its group has order two on a repeat of two — the whole of it. Any figure with plain weave on one side of the boundary therefore has 4 ÷ 2 = two relative origins, and one of those is the reflection of the other. There is almost nothing to choose, which is why a satin stripe on a plain ground has never produced a surprise and a damask has.

Three consequences follow, and the third is the practical one.

The effect is a property of high-order weaves. The more symmetric a weave is, the fewer distinct ways it can be phased against a partner, and the plain weave is the most symmetric there is. The pairs that can go wrong are the pairs with long repeats, which is exactly the family a jacquard uses.

Two weaves that share a symmetry are cheaper to test than two that do not. The census cost is the number of origins times 2^(k²), so a pair whose groups intersect trivially costs r² censuses and a pair like the twills above costs four. A designer testing a new pair should compute the intersection first.

And the phase has to be recorded. Sequence, step and hand fully specify a twill, and a figured cloth needs a fourth item: the offset of the ground against the figure. Point paper does not carry it, a manual does not print it, and a design handed on as two drafts plus a profile is a design that cannot be rebuilt safely — the person rebuilding it will start each weave where their pencil went down, and the arithmetic above says that is a one-in-two chance of the wrong answer on the pair this essay found.

Where the model stops #

The block rule is stated for aligned blocks and nothing else. A real jacquard figure is a region drawn freehand on squared paper, and its boundary is a staircase whose treads are not all the same size. Everything here is about a boundary that lies on a lattice; a boundary that does not is a superset of these cases and can only be worse.

Rectangular blocks are not covered. A block that is a repeat wide and half a repeat deep satisfies half of the condition, and the proof needs both halves. Whether the weaker condition is enough is not known here and the census has not been run.

Nothing here is about how the boundary looks. A figure that passes every test on this page can still show a hard line where the two weaves meet, because two weaves with different interlacing rates take up thread differently and a boundary between them is a place where the cloth is not flat. That is the same missing piece a striped cloth’s join has, and it needs two reed pitches and a model of take-up that this site does not have.

And the criterion is still the criterion. It answers whether the fabric separates, exactly, and says nothing about friction, fibre or finish. A milled figured cloth whose two layers are held together by felted surface fibre will read as two here and behave as one, which is the standing limit recorded from the other side.

The generalisation #

Strip the cloth out and what is left is a statement about combining two periodic structures over a region, and the shape of it is not special to weaving.

A property that is preserved by taking unions along one axis need not be preserved by taking unions over a region, because the one-dimensional case has a shared boundary of full measure and the two-dimensional case does not. That is the whole of the block rule and the whole of its failure. And the phase effect is the second half of the same observation: when two periodic objects are combined, their relative offset is a genuine parameter of the combination even when it is a gauge freedom of each one alone.

The practical form is worth stating because it inverts a piece of common sense. Finer detail is normally a design luxury, paid for in machine cost. Here it is a structural risk, paid for in a property nothing on the drawing shows — and the coarser design, the one that looks cruder, is the one that cannot fail.

Who found it, and when #

Block design is old and thoroughly worked out in practice. Summer and winter, huckaback, M’s and O’s and the crackle weaves are all designed on profile drafts, and the manuals give the rule for stepping a figure boundary in terms of the ground weave’s repeat. The rule is right. What the manuals give as its reason is that a shorter step produces long floats at the boundary — which is true, is the commonest failure by a factor of thirty, and is not the only one.

The connectivity criterion is the periodic-fabric work of the 1980s, Grünbaum and Shephard’s, and it is a theory of a single periodic fabric. Two fabrics blended over a region is not an object in it, so the question this essay asks does not arise there and there is nothing to have missed.

The relative-origin result is this site’s, and it exists because the census was run at every origin rather than at the one the drafts happened to be written at. That was not foresight; the sweep was added to check that the phase had been divided out properly, on the assumption that it had been. It had not.

Where the ladder goes next #

The next rung takes the same block designs and asks what they cost. A figure’s shaft count turns out to depend only on how many distinct block-columns it has, so a design a hundred and twenty ends wide costs exactly what one twenty-four ends wide costs if it repeats the same columns — and a genuinely new block-column costs exactly one repeat of shafts, every time.

After that the boundary itself: a woven outline is a staircase whose tread is a block, and a block is at least one repeat, so a figured cloth has a resolution that a finer machine cannot improve.

Sideways, the same pair of weaves in one cloth is what a striped cloth and a check are, one dimension down, and what a damask is with the two weaves chosen to be each other’s reverse. Further out, the same question asked of a stack rather than a region is the double cloth, where two cloths are the intention and one of them being accidental is the failure.