What else the relative origin decides

A figure is not a stripe found a quantity nobody had named. Two weaves blended over a region have a relative origin — where one starts with respect to the other — and it is the only part of the pair of origins that cannot be slid away, because point paper has no origin for a weave taken alone and does have one for a pair taken together.

That rung found the origin decides whether the cloth holds together: eight of sixteen origins safe at every figure there is, eight breaking on more than a third of them. Its companion rung found a second consequence, in how evenly the loom works the warp.

Its own list of what was not done says the rest: the relative origin has two known consequences and there is no reason to think it has two. Sweeping it against every other quantity the site already computes is one loop and has not been done.

The claim #

The relative origin decides at least three things, and they are not decided together.

The layer count — the previous rung’s finding. The longest float, which no account of figured cloth mentions and which is the property a weaver inspects a cloth for. And the interlacing rate, which decides firmness, drape, abrasion and how closely the cloth can be set.

They are not decided together: the sixty-four origins fall into three different partitions, one per moving quantity, so a designer choosing an origin for one reason is choosing the others independently and by accident.

There is also a fourth thing this rung found, and it is a defect rather than a result. A caption on this site says the opposite, and it is wrong.

The sweep #

The method is the loop the previous rung said had not been run. Hold the figure weave, the ground weave, the profile and the block; slide the ground through every relative origin its own repeat admits; and compute, at each, every quantity this site knows how to take off a matrix.

Six of them: the layer count, the longest float, the interlacing rate, the shaft count, the number of distinct lifts, and the weft-face fraction. Each is a measure some essay here already argues about. None is a measure of the boundary.

For an eight-end satin on its own reverse at a block of two, over all sixty-four origins:

The weft-face fraction is in the list precisely because it was expected not to move, and a column that cannot move is what makes a count of moving columns a count rather than a total. It moves — because the profile covers only part of a repeat of each weave, so sliding the ground changes which cells the region happens to catch.

Three partitions, not one #

The interesting question is not whether a quantity moves but whether two quantities move together. The interlacing rate decides a cloth’s firmness and the float decides most of what a reader sees; if those two moved in step, a designer choosing one would be choosing the other. If the three moving columns partitioned the sixty-four origins the same way, the whole effect would be one number wearing three hats. They do not.

Group the origins by the value each column takes and compare the groupings: three columns, three distinct partitions. The origin is three decisions, not one.

That matters practically. The previous rung’s advice — sweep the origins and pick one that is safe — turns out to be advice about one axis of a three-axis choice, and the safe origins are not the ones with the shortest floats and are not the ones with the highest interlacing rate.

The float, which is the one a weaver looks at #

Of the three, the float is the one that matters most in the loom room, and it is the one nobody could have predicted from the previous rung’s framing.

The framing was: a weave started somewhere else is the same cloth, which is true — floats, interlacings, balance, shafts and layers are all unchanged by sliding a weave along. Every one of those statements is about one weave. None of them survives into the composite, because the composite is not a translate of anything: the boundary stays where it is while the ground slides underneath it, so the cells the boundary catches are different cells.

For the chequer profile above, the longest float is three at two origins and eight at the other two. Three is an ordinary 2/2 twill float and eight is a fault — the kind of length a float limit exists to exclude, arriving through a door no float limit is watching. A weaver looking at the four drafts on point paper would see two acceptable cloths and two they would send back, and nothing in either weave’s own reading distinguishes them.

That is a stronger practical statement than the integrity finding, because integrity is invisible and needs a computation while a float of eight is visible and needs an eye. The origin was already being chosen, by inspection, on a criterion nobody knew they were applying.

The caption that was wrong #

This rung found the finding by asking a question of the machinery, and the machinery answered by contradicting a caption on this site.

The figure of four origins on the rung below carries a description that ends: the four cloths have the same floats, the same interlacings, the same shafts and the same layer count, and nothing on point paper records the offset.

The last clause is right. The rest is not. For the pair and profile that figure draws, the four cloths have longest floats of 3, 3, 8 and 8; layer counts of 1, 1, 16 and 9; interlacing rates of 0.50, 0.50, 0.50 and 0.25; and shaft counts of 4, 4, 2 and 3. Two of the four panels are cloths in nine and sixteen pieces — the failure the whole criterion exists to catch, and the caption says they are the same as the other two in every measure.

It is worth being precise about how that happened, because the mechanism is more interesting than the mistake. The claim was true of the thing the essay was arguing about — the two weaves, which really are unchanged by sliding — and was written into a caption about the four composites, which are not. The two sentences differ by one word and the word is invisible in context.

There is also a machinery reason it was never caught. The function that builds a figured cloth reported its interlacing rate by reading a field the interlacing function does not have, so the value was undefined on every figured cloth ever built. Nothing printed it, so nothing failed. It was found by this rung’s sweep asking whether that column moves, which is the first time anything had read it.

Both are fixed at the source: the caption now measures the four panels and says what they share and what they do not, and the interlacing field reads the right name. The general shape is one this site has recorded before in other clothes — an assertion written against the thing in front of the writer rather than against the claim being made.

What was counted, and how #

The sweep is exhaustive over the ground weave’s own repeat: every offset in both directions, sixty-four for an eight-end weave and sixteen for a four-end one. For each, the composite is built at full detail — floats, shafts and interlacings computed rather than skipped, which is the expensive path and is the point.

The partition comparison is done by grouping origins by value and comparing the groupings as sets, rather than by comparing correlations. Two columns that happen to take the same number of values are not the same decision, and a correlation would be a summary where an exact comparison is available.

The machinery asserts that the sweep visited every origin the ground’s repeat admits, which is the kind of off-by-one that would quietly halve a result.

What is not in the sweep is the census over all 65,536 profiles at each origin — the previous rung’s own computation. That is a seventh column, it is far more expensive than the other six together, and it is reported separately rather than folded in, because tying the two together in one table would suggest they were computed the same way.

Where the model stops #

One profile at a time. The six columns are computed for a particular figured cloth, so they are statements about that design at each origin rather than about the pair of weaves. A different profile gives a different table, and whether the three partitions are properties of the pair or of the profile has not been established.

Three moving columns is a lower bound. The six quantities are the ones this site can compute. A cloth has properties this site cannot compute — its thickness, its drape, its air permeability — and there is no reason to think the origin leaves them alone.

The partitions are not characterised. The previous rung found that for a step-one twill the sixteen origins collapse to four classes, because such a twill is invariant under sliding one end and one pick together. Nothing similar has been worked out for the partitions here; they are reported as counts rather than as structures.

And the two known consequences are still only checked at one pair. The integrity split and the flex split are cleanest for the 2/2 and 3/1 twills, which is where both were found; whether the three new partitions line up with either has not been checked at another pair, and that shortfall now stands at three sweeps rather than two.

Why three columns move and three do not #

The six quantities split cleanly into those the origin touches and those it cannot, and the division is not arbitrary — it is a division between two kinds of measurement.

The three that do not move are counts over the whole repeat. The shaft count is the number of distinct columns in the composite, the lift count the number of distinct rows, and the layer count in this particular case is one everywhere. Sliding the ground permutes which cells of the ground weave land in which part of the region, and a count over the whole matrix does not care about a permutation of that kind so long as the region’s shape is unchanged.

The three that move are local. The longest float is a maximum over runs, so it is decided by whichever single run happens to be longest — and a run that straddles the boundary is built from a piece of one weave and a piece of the other, so its length depends entirely on where the two happen to meet. The interlacing rate and the weft-face fraction are averages over the region, and the region is a part of a repeat rather than a whole one, so the cells it catches change as the ground slides.

That gives a test worth carrying past this pair. A quantity that is a sum or a count over a complete repeat is very likely origin-invariant; a quantity that is a maximum, or an average over a sub-region, very likely is not. The one exception in the table is the layer count, which is a global property and which the previous rung showed moving dramatically at other origins and profiles — because connectivity is not a count over cells at all but a question about a graph, and a graph can be disconnected by one edge going missing anywhere.

The rule has one more use, and it is about what has not been swept. A cloth’s thickness, its drape and its air permeability are all averages over a region rather than counts over a repeat, so the category test predicts that the origin moves every one of them — which is a sharper version of the shortfall recorded below, since it says not merely that more consequences may exist but which quantities to look at first.

Predicting where a sweep will find something is not the same as running it, and none of those three has been run here.

So the sweep’s three moving columns are not three unrelated surprises. Two of them move because the region is not a whole repeat, one moves because a maximum is a local statistic, and the layer count moves because connectivity is fragile in a way no cell count is. Knowing which category a quantity is in says whether to expect it to move before the loop is run — which is the cheapest form the finding takes, and the form that would have prevented the caption.

The generalisation #

The methodological point is the one worth extracting, and it is uncomfortable.

Finding two consequences of a parameter is not evidence that it has two. It is evidence that the search looked twice. The previous rung found integrity because it was looking for integrity, and found the flex spread because a second ladder happened to meet the same number — which is a coincidence of attention rather than a survey. The only way to know how many consequences a parameter has is to sweep it against everything computable and count the columns that move, and that is a cheap loop wherever the quantities are already implemented.

The second point is about invariance and where it lives. A symmetry of the parts is not a symmetry of the whole when the whole has a fixture in it. Sliding a weave is a symmetry of that weave. Sliding it inside a figure is not a symmetry of the figure, because the boundary does not slide — and every invariant of the weave that gets quoted about the composite is a claim that has to be re-established rather than inherited. This site made that error in a caption and it is an easy one to make, because the two sentences look alike.

Who found it, and when #

The relative origin is the previous rung’s finding and is, as far as this site can tell, not in the literature: point paper has no origin, weaves are catalogued modulo translation, and the object that has an origin — a pair of weaves in one cloth — is not an object the periodic-fabric theory treats.

Nothing here is a discovery about weaving so much as a survey of one that was already made. What it adds is that the parameter is not a specialist’s parameter about a computation nobody runs: it moves the longest float, which is the oldest and most practical measure there is, and which weavers have been rejecting cloths on since long before anybody could compute anything.

The defect it turned up is this site’s own, and recording it is the point of writing down what an argument found rather than only what it built.

Where the ladder goes next #

The next thing this ladder owes is the characterisation of the partitions — the previous rung showed a step-one twill’s sixteen origins collapse to four for integrity, and the same question for the float and the interlacing rate is open.

Sideways, the block’s own shape decides things too, and the two rungs have never been swept together: origin against block shape is a two-dimensional sweep and neither axis has been held against the other.

Further out is the question the whole ladder keeps arriving at. Two weaves with different interlacing rates take up thread differently, so a figure boundary is a place where the cloth is not flat — the same missing piece a striped cloth’s join has — and the origin moves the interlacing rate, so the origin moves the ridge. That is a physical consequence rather than a combinatorial one, and it needs the take-up model this site does not have.