A figure is harder on its warp

Two of the ladders here were built for unrelated reasons. One is about a figure — one weave inside a region of the point paper and another outside it — and what that does to the cloth’s integrity. The other is about the shed, and how much work the loom does on each warp end.

They meet at a number neither was looking for.

The shed ladder found that every shift-rule weave — one whose columns are all rotations of one column, which is plain, every twill, every satin and every sateen — flexes all its ends exactly as often as each other. Rotating a column does not change how many times it changes value, so the loom’s work is spread perfectly evenly across the warp, whatever the float length. That covers essentially the whole traditional catalogue of basic weaves, and it is a one-line theorem.

Figure one shift-rule weave on another and the evenness can go. Whether it does is decided by the relative origin — where the ground weave starts against the figure — which is the same quantity the figure ladder found deciding whether the cloth is one cloth at all.

Why figuring can break the evenness #

An end’s flex count is the number of times its column changes value round the repeat. In a figured cloth that column is a stack of segments, one per block, each following whichever weave the profile put there.

Inside a segment the changes are the weave’s own and every end sees the same number. At a block boundary the change depends on the phase — whether the last pick of one segment and the first of the next happen to agree — and that is decided by where the end sits in each weave’s column cycle.

Two ends in the same block-column with different phases therefore pick up different numbers of boundary changes, and the flat row becomes a ragged one. Nothing about either weave has changed; what has changed is that a boundary now exists, and the boundary falls at a different place in each end’s column cycle.

Where the two ladders meet #

Take a check profile — four blocks, two figure and two ground — in a 2/2 twill on a 3/1 twill, at a block of four. Four ends is a whole repeat of both weaves, so this is a coarse figure: by the block rule it cannot separate, at any relative origin, and it does not. All sixteen origins give one cloth.

Sweep the relative origin anyway and look at the flex counts.

Eight of the sixteen give an even cloth: every end flexes eight times. The other eight give a spread of six to ten.

The split is exact and it collapses to one number. A 2/2 twill is unchanged by sliding one end and one pick together, so only the difference of the two offsets matters; it takes four values; and two of those four are even. Which is precisely the structure the figure ladder found for integrity at a finer block — the same one-number offset, splitting the same four ways.

What that means, and what it does not #

It does not mean that the two ladders’ findings are the same finding. They are two different consequences of one parameter, asked at two different block sizes.

At a fine block the offset decides integrity: eight origins are safe at every profile and eight break on more than a third of them. At a coarse block integrity is settled — the block rule guarantees one cloth at every origin — and the offset has stopped mattering for that. It has not stopped mattering. It now decides how unevenly the warp is worked, and it splits the origins the same way.

So the honest statement is that a relative origin is a real parameter of a figured cloth, and its consequences do not stop at the one that was noticed first. A designer who steps a figure coarsely enough to be structurally safe has not disposed of the offset; they have moved it from one consequence to another.

The practical form is small and specific. Two mills weaving the same figured cloth from the same specification, whose designers happened to start the ground weave at different places on the point paper, will get identical fabric and different warp-break rates.

The product, once more #

The previous rung’s quantity was the flex count multiplied by the strain of the end’s own shaft, which is the only number on this site that is a function of both the matrix and the machine. A figured cloth makes it worse in both factors at once.

The flex count is now uneven, as above. And the shaft count is high — a figure costs the number of distinct block-columns times the repeat — so the harness is deep, and a deep harness has a large strain ratio between its front and its back. On a twenty-four-shaft harness that ratio is 3.75.

So a figured cloth spreads a ragged flex count across a harness with a wide strain range, and the product’s spread is larger than either factor’s. On the damask check the work spread across one repeat is 4.09 to one, and on a diagonal figure of the same two weaves it is 7.51.

The easiest weave on a warp is worth putting at the end of the sequence, because it is where the spread the figure pays for is smallest.

What was counted, and how #

The flex count is computed per column, cyclically, from the composite matrix that draws the figure. The composite is built the same way the figure ladder builds it: both weaves tiled from a common origin, the profile choosing between them cell by cell.

One step is checked rather than assumed and it is the step that lets a per-end quantity be reported per shaft: every end on one shaft has the same column by construction, so it must have the same flex count, and that is asserted end by end. In a figured cloth the shaft count is large and the check has more to do than in a plain weave.

The origin sweep runs the whole flex computation at each of the sixteen relative origins and classifies each as even or not. The two ladders’ results are then compared, and it is worth being explicit that the comparison is not an assertion: they are computed independently at different block sizes and the observation that both split eight and eight is reported rather than required. An assertion tying them would be asserting a coincidence.

The census over eleven weaves runs both halves of the shift-rule theorem. Every shift-rule weave must come out even, and some weave that is not one must not — and the figured entries are in the list precisely because they are figures of shift-rule weaves that are not even, which is the essay’s content sitting inside its own check.

Why eight and eight, and what the split is made of #

The sixteen origins dividing exactly in half is a clean number and it is worth taking apart, because the halving is a property of the two weaves rather than of the figure.

A relative origin is a pair — how many ends and how many picks the ground has been slid by — so there are sixteen of them on a four-end repeat. A 2/2 twill is unchanged by sliding one end and one pick together, which is what makes it a twill, so the sixteen collapse to four classes indexed by the difference of the two offsets. The same is true of the 3/1 ground.

So the whole sweep has four distinct answers, each occurring four times, and the question is how many of the four are even. Two are: the offsets at which the block boundary falls where the two weaves’ columns happen to agree in parity, which for a pair of four-end shift-rule weaves is exactly half the difference classes.

Four classes, two even, four origins each — eight and eight. The halving is arithmetic and would be a different fraction for a different pair: an eight-end satin on its own reverse has sixty-four origins collapsing to eight classes, and how many of those are even is a question about eight rather than about four.

That says the correspondence with the integrity sweep is a correspondence of structure rather than of value. Both quantities depend on the offset only through the difference, because both are computed from a composite of two shift-rule weaves and both inherit the same collapse — so both split the sixteen into four groups of four, and how those four groups fall is a separate question for each. They agree on the shape and not necessarily on the membership, which is a weaker and more defensible statement than the coincidence of two eights suggests, and it is the one the essay’s own refusal to assert the correspondence was protecting.

There is a practical reading of the collapse as well. Only the difference of the two offsets matters, so the parameter a designer is unknowingly choosing has four values rather than sixteen — which makes it a far more tractable thing to sweep than it first appears. A drawing office that wanted to check a figured design against both consequences would have four composites to build rather than sixteen, and the four are generated by sliding the ground one pick at a time.

The collapse is a property of the two weaves rather than of the design, so it holds for every profile on that pair.

Where this would show up #

A finding that turns on an offset nobody records is worth asking about from the other direction: if it is real, where would anybody have met it?

The place is a mill weaving somebody else’s design. A figured cloth is specified by its profile, its two weaves and its sett, and every one of those is written down. The relative origin is not, because it is not a property of any of them — it is a property of how the composite draft happened to be assembled, which is done by software from the specification and is not a decision anybody records making.

So two mills weaving the same specification can assemble the same composite at different offsets, produce fabric that is identical in every measurable respect, and see different warp-break rates. The obvious diagnoses would all be wrong: the yarn is the same, the sett is the same, the loom is the same kind of loom, and the drawdowns match square for square when they are compared, because comparing drawdowns means comparing the cloth and the cloth is the same.

That is a testable prediction and this site is in no position to test it. What can be said is that the mechanism is exact, the size of it is a factor across the ends of one repeat, and the parameter it depends on is one nobody has any reason to have written down. A discrepancy of that shape would be very hard to trace and very easy to attribute to something else, which is the honest reason to expect it has been met and not identified rather than to expect it has never happened.

Where the model stops #

A flex count is not an abrasion. How much damage a passage through a heddle eye does depends on the eye’s shape and finish, the yarn’s hairiness, the size on it and the tension — none of which is here. The count is a load index, and what it is good for is the ratio between ends.

The threading is taken as its shaft count requires. A figured cloth can be threaded many ways, and a threading chosen to put the most-flexed ends on the front shafts would reduce the product’s spread substantially. Nothing here optimises that, and it is a real opportunity rather than a hypothetical one — the allocation problem is the same one where the heddles go solves for a different objective.

The correspondence between the two ladders is at one pair of weaves. The 2/2 and 3/1 twills are the pair where both effects are cleanest, because the twill’s invariance under a diagonal slide collapses sixteen origins to four. Another pair has a different collapse and the correspondence has not been checked there.

And nothing here says a figured cloth breaks more ends. It says the loom’s work is distributed less evenly, which is a statement about a computed index and not about a mill’s stop rate. A real warp’s breaks depend on the yarn far more than on any of this.

The generalisation #

The move worth carrying is the one this essay is: when an invisible parameter is found to control one thing, look for what else it controls.

A relative origin is a gauge freedom of each weave separately and a real quantity of the pair. That was the figure ladder’s finding, and the natural reading of it is that the parameter matters because it can break the cloth — which makes it a structural concern with a threshold, and makes stepping a figure coarsely enough the end of the matter.

It is not the end of the matter, because the parameter did not exist in order to break the cloth. It exists because two periodic things were combined, and every quantity of the combination is a function of it. Integrity is the first consequence anybody found because it is the loudest. The loom’s load is a second. There is no reason to think it is the last.

The diagnostic that produces this is cheap: having found that a quantity depends on a hidden parameter, sweep the parameter against every other quantity already computed. Here that was one loop over sixteen values of an offset and a function that already existed.

Who found it, and when #

The shift-rule evenness is a one-line observation and is this site’s own. Nobody in the trade needs it, because everybody knows a plain weave is hard on a warp and a satin is easy, and the flat-across-the-warp part is invisible when every weave anybody uses has it.

That figured cloths break ends unevenly is thoroughly known in practice. A weaver watching a jacquard knows which parts of the design cause trouble, and the remedies — easing the tension, splitting the warp onto two beams, putting the hardest-worked ends on their own beam — all exist because the problem does.

What appears to be new is the cause identified here, and it is not the one anybody would reach for. The obvious cause of uneven work in a figured cloth is that the figure and the ground interlace at different rates, which is true and is a large effect. The offset is a second and much stranger one: two cloths with the same two weaves and the same profile, differing only in where a pencil went down, differing in warp load by a factor.

Where the ladder goes next #

This closes the shed ladder at three rungs and leaves the same thing open all three of them leave: a force. Every number here is a strain or a count, and what either costs in newtons needs a yarn model this site does not have.

What it opens is smaller and more tractable: a threading chosen to level the product rather than to be convenient. The allocation machinery already exists for the heddle-load problem, the objective is different, and the two objectives are very likely in conflict — which is the kind of question a site with both quantities computed is in a position to answer.

Sideways, the parameter this essay shares with the pattern field is the relative origin; the two factors of the product are the interlacing count and the shed’s own strain; and what a figured cloth costs in shafts, which is what makes the harness deep, is in the pattern field.