A stripe is a partition of the warp

A striped cloth arrives in a mill as an instruction that sounds like decoration: twelve ends of plain, eight of satin, repeat across the width. Nothing about that phrasing suggests a structural question, and the loom does not care how the instruction was phrased. What goes on the beam is a single warp, and what comes off the loom is a single fabric with one repeat, one shaft count and one answer to whether it holds together.

That matters because almost everything this site measures is a property of the whole repeat. The draft is a matrix, and a striped draft is a different matrix from either of the weaves that went into it. Its floats are theirs, its interlacings are theirs, and its shaft count and its layer count are not.

The partition, and what it does to the repeat #

A stripe is a partition of the ends. So many ends belong to the first rule, so many to the second, and every pick crosses all of them. That is the whole of the construction, and cloth.js’s stripes does exactly that — takes a list of bands, tiles each weave across its own ends, and returns one matrix.

The first consequence is arithmetic and is felt in the pattern chain rather than in the cloth. Each band’s weave repeats after its own number of picks, and the striped cloth cannot repeat until all of them do, so the pick repeat of a stripe is the least common multiple of its parts’. Plain beside a 2/2 twill repeats after four picks. Plain beside an eight-end satin repeats after eight. Plain beside a five-end satin repeats after ten, and a cloth striping all ten of the weaves in this essay repeats after 120 — which is not a notional number but the length of chain the loom has to store.

The second consequence is the whole essay: the band is not the weave. Two ends of a 2/2 twill are two columns of it, not a 2/2 twill, and a band eleven ends wide of an eight-end satin is eight columns plus three of them again. Almost every interesting thing here follows from taking that seriously.

The harness cost is a union #

A shaft is a distinct column. That is not a convention; it is what a shaft is, because every end whose column is identical must lift on exactly the same picks and may therefore hang on the same heddle bar.

So the shaft count of a striped draft is the size of the union of its bands’ column sets — and a union is bounded on both sides. It is at least the larger of the two sets, because that set is inside it, and at most their sum, because nothing else can be. The trade rule of thumb is the upper bound: a satin stripe on a plain ground needs ten shafts, eight and two.

The rule of thumb is a sum where the truth is a union, and the difference is exactly the number of columns the two weaves have in common. Nobody asks that question, because nobody has any reason to expect the answer to be anything but zero.

Forty-two of forty-five share nothing #

Running the union over every pair gives a table with almost no structure in it and three cells that have all of it.

Three pairs of forty-five share a column. The 2/2 twill and the 2/2 basket, the 2/2 twill and its own herringbone, and the 2/2 basket and that herringbone. Everything else in the table — plain against every twill, every twill against every satin, the two satins against each other, the 3/3 basket against the 2/2 twill it looks so like — shares nothing at all, and a stripe of the two costs exactly the sum.

The three that do share are worth stating individually because each is a different kind of saving. A 2/2 twill beside its own herringbone costs four shafts and not eight: a herringbone reverses the threading and not the weave, so its columns are the twill’s columns in a different order and its column set is identical. A 2/2 basket beside a 2/2 twill costs four and not six, because the basket’s two columns are two of the twill’s four. And the basket beside the herringbone costs four for the same reason at one remove.

In none of the three is the overlap partial. Where two of these weaves share anything, the smaller column set lies wholly inside the larger. That is not luck. A weave built by a shift rule — every twill, every satin, every basket — has for its column set every rotation of one column down the picks, and two such sets are either identical or disjoint — a column belonging to both would drag all of its rotations in with it. A partial overlap would mean one of these weaves was not built by the rule it claims.

The other twill, which shares nothing #

The pair a reader expects to share is the 2/2 twill and the 3/1 twill, and it does not.

Both repeat on four ends, both need four shafts, both are drawn by shifting a column of four squares one end at a time, and their diagonals run the same way. A stripe of the two costs eight shafts, which is every shaft on an ordinary dobby before any other weave has been considered. The column of a 2/2 twill has two filled squares and the column of a 3/1 twill has three, so no rotation of one is ever the other, and the two sets of columns could not meet however wide the cloth.

That is the practical form of the finding. Sharing depends on the shape of one column, not on the family the weave belongs to — and family resemblance is exactly what a designer reaches for when guessing which stripes will be cheap. The 2/2 basket, which nobody would call a twill, is free beside one; the 3/1 twill, which is a twill in every book, is not.

What was counted, and how #

Both halves of this essay are enumerations rather than examples, and both are run while the figures are drawn.

The cost table is computed rather than compiled. Ten weaves, each generated from its own rule; every column written out at the common pick repeat of 120 and stored as a string; the union taken pair by pair. The comparison is also made at 240 picks and required to give the same answer, because a set comparison that quietly depended on how far the repeat was written out would be the same defect the repeat lattice was written to expose. Forty-five pairs, three sharing, and an assertion that refuses a partial overlap.

The integrity census is stripeIntegrityCensus. Every ordered pair of the ten weaves at ten band widths each — one, two, three, four, five, six, eight, ten, twelve and sixteen ends — which is 9,000 striped drafts. Each is built as one matrix and handed to layers, and each band is separately built as the draft it actually is and handed to layers as well, so the stripe’s answer and its parts’ answers can be compared.

The result is the sharpest thing in the essay.

606 of the 9,000 describe more than one cloth. And not one of the 606 has both its bands sound. Every single failure has a band cut narrower than that band’s own weave repeat, and there are no failures at all among the 4,648 stripes whose two bands are each one cloth on their own.

Why striping cannot break a sound cloth #

That is a theorem and the census is its check, which is the right way round.

The integrity criterion is a question about a directed graph. Every strand — each end, each pick — is a vertex, and an edge runs from the strand underneath to the strand on top at every intersection, because putting the lower one in the lifting set forces the upper one in too. The fabric is one fabric exactly when that digraph is strongly connected.

Now take a stripe. Restrict the digraph to one band’s ends together with all the picks, and what is left is precisely that band’s own digraph, because every intersection between those ends and those picks is in both. So if the band is one cloth, that subgraph is strongly connected. The same holds for the other band. And the two subgraphs share every pick in the cloth.

Two strongly connected subgraphs sharing a vertex are strongly connected. So a stripe of sound bands is sound, whatever the weaves and whatever the widths, and the 4,648 opportunities the census gave the claim to fail are 4,648 confirmations.

How narrow a band may be #

The census turns around into a number a weaver can use. If failure only ever comes from a band cut too narrow, the question is how narrow is too narrow, and the answer is not the weave’s repeat.

An eight-end satin needs its full eight ends before a band of it is one cloth. A five-end satin needs five, a 3/1 twill needs four. But a 2/2 twill is one cloth at three ends of its four, a 2/2 basket at three of its four, and a 2/2 herringbone at three of its eight. A band narrower than that is not a fabric even in principle, and only becomes one when something is put beside it.

Which raises the other half. A band that cannot stand alone is not doomed: of the 4,352 stripes with at least one unsound band, 3,746 come out as one cloth anyway, held together by the band beside them. Only 606 do not. So the narrow band is a risk rather than a rule, and it is a risk no drawing declares.

Why the three that share are the only three that could #

The census reports three sharing pairs out of forty-five and calls the shift-rule argument in to explain why an overlap is never partial. The same argument, taken one step further, says which pairs can share at all — and the answer is a decision procedure a designer could run on the back of a docket.

A weave built by a shift rule is generated from one column word and a shift. Write the column as a string of ups and downs down the picks; move to the next end by rotating that string by the shift; the weave’s column set is the orbit of the word under repeated rotation by that amount.

Two facts follow immediately.

Two weaves from different words never share. An orbit under rotation contains only rotations of its own word, so a column belonging to two orbits would make the two words rotations of each other — and then the weaves are the same weave written from a different starting pick. A 2/2 twill’s word has two ups and a 3/1 twill’s has three, so no rotation carries one to the other and the two sets are disjoint however wide the cloth. That is the “other twill” the essay above finds surprising, and the surprise is entirely in the family name.

Two weaves from the same word share according to their shifts. Rotating by one generates the whole orbit; rotating by two generates half of it; rotating by four generates a single column. So the column sets of one word at different shifts are nested — each a subgroup’s orbit inside the full one — which is precisely why every overlap in the table is total rather than partial, and why the smaller set is always the one inside.

All three sharing pairs are that second case and nothing else is. A 2/2 basket is the 2/2 twill’s word at a shift of two, so its two columns sit inside the twill’s four. A herringbone is the twill’s word at a shift of one and then minus one, which reorders the orbit and does not leave it, so its set is identical. And the basket beside the herringbone is the first relation composed with the second.

The procedure a designer wants is therefore two comparisons rather than a table:

Write one column of each weave down the common pick repeat. If the two strings are not rotations of one another, the stripe costs the sum. If they are, the cheaper weave’s shafts are already on the loom and the stripe costs the larger of the two alone.

That also says how much of the table could ever have been different. Of the ten weaves here, the ones sharing a word are the 2/2 twill, its basket and its herringbone — three weaves, three pairs, and the count is exactly what the enumeration found. A fourth sharing pair would have required a fourth weave built on the same word, which is why adding a 4/4 basket to the list would find one and adding another satin would not.

Where the model stops #

The matrix knows nothing about the boundary as a physical place. Two bands with very different interlacing rates take up thread at different rates, and a cloth woven from one beam holds them all at one tension, so a striped fabric puckers along its joins in a way the relief weaves exploit deliberately. The integrity criterion has no opinion about that at all: a stripe that passes it can still be an unsatisfactory cloth.

The setting is quoted once and the cloth wants two. The weave decides the sett, and a stripe has two weaves, so the plain ground and the satin band want different reed pitches and are given one. That shows in the finished cloth as a difference in width contraction between bands, and nothing here computes it.

The census is ten weaves and ten widths. The theorem is general and the counts are not. A different list of weaves would give different totals, and the three sharing pairs are three pairs of these ten — a list including a 4/4 basket would find a fourth.

And the shaft count is not the harness. A shaft is a distinct column and says nothing about how many heddles hang on it, which for a narrow stripe on a broad ground is where the difficulty actually is.

Who found it, and when #

The stripe is older than any of the machinery used to describe it. A warp with two weaves in it needs nothing but a threading that changes part way across, which is available on the simplest loom there is, and striped cloths are among the oldest patterned fabrics known.

What the trade has always had is the rule of thumb: add the shafts. It appears in that form in every draughting manual, and it is right in forty-two cases out of forty-five here because the exceptions are exactly the cases nobody thinks to check — a weave beside its own reversal, or beside a doubling of itself. Both are pairs a designer would reach for on grounds of appearance rather than economy, which is a pleasant accident and not a plan.

The column-set framing is not a textile idea; it is what falls out of writing the draft as a matrix and asking which columns are equal, and it belongs with the harness census and what a dobby stores. The integrity theorem is graph theory of the least demanding kind, and it is worth stating precisely because the intuition runs the other way: a striped cloth looks more fragile than a plain one, and it is not.

The one thing here the trade genuinely does know and does not write down is the narrow band. Weavers do not put a single end of satin in a plain ground, and the reason given is that it looks wrong. The arithmetic says it also is not cloth on its own, and only survives because of what is beside it.

Where the ladder goes next #

The next rung takes the same partition and applies it to colour rather than to weave: a check is two stripes and a tartan is one, where the bands are colour orders and the join is where the eye actually looks.

Beside it sits the partition turned inward — a cord is a stripe with no colour in it, where the groups are two threads wide rather than twelve and act as a single coarse thread.

What the pictures here cannot show. Every figure is a matrix drawn on point paper, so the join is a ruled line and in cloth it is a place where two fabrics of different take-up meet. The drawings say exactly which shafts are needed and exactly how many cloths the draft describes, and say nothing whatever about whether the finished stripe lies flat.