An eight-end shading is pinned at three everywhere but its centre

An even shading cannot keep its surface level pressed every six-end shading whose middle tones float no more than two and found them all sinking to one depth, 43.3 micrometres on a sheeting. The uniformity had a reason: a float limit is a run-length limit, a run-length limit forces a floor on how often a thread changes face, and at six ends the limit of two pinned the midtone exactly at its floor.

It ended on the next repeat. At eight ends the floor says every even shading sinks at least 42.8 micrometres, and the question was whether any reaches that floor, and whether they all sink to one depth or spread out above it. The squares of order eight are far too many to walk one at a time, and the essay proposed a search that builds a shading tone by tone and abandons any partial one that already floats too far.

The search has been run. Its first finding is that the question, as put, has no chain to ask it of.

Float two is out of reach at the second tone #

A tone of k marks in every pick and every end, drawn from an eight-end repeat, has 8 − k unmarked places in each pick, and those are the weft floats on the face. At the second tone there are two marks and six unmarked places, and two marks can divide a closed pick into at most two runs. Six places in two runs means a run of at least three.

So no eight-end shading holds its second tone to a float of two. The even family the six-end census was built on — every middle tone held to two — does not exist at eight ends at all. And since the float decides a cloth’s lustre, its snagging and how loosely it holds a thread, the second tone of every eight-end shading carries a float of three into all three whether the designer wants it or not.

And a float of three at the second tone is reachable in exactly one way. Six unmarked places in two runs of at most three means two runs of exactly three, which means the second mark of every pick sits exactly opposite the first, four ends away. The second part of the shading is forced by the first.

And the third mark cannot undo it #

The third tone adds one mark to every pick. With the first two opposite — which is what holding the second tone to its least float requires — the pick’s unmarked places are two runs of three, and the new mark falls in one of them.

It splits that run. It does nothing to the other. So the pick still floats three, and every pick of the third tone floats three, and so does the third tone as a whole. The float of two that the third tone’s own counting would allow — five unmarked places in three runs — needs three runs, and a pick that already has two runs of three can only be given a third by breaking one of them, which leaves the other intact.

A shading could buy a float of two at its third tone only by letting the second float four, which is no bargain for a family defined by its longest float. The sixth and fifth tones are the second and third read from the other face — their unmarked cells, taken in reverse order, are a shading of their own, with the ends playing the part the picks played. So in the tightest even family tones 2/8, 3/8, 5/8 and 6/8 are pinned at a float of three, by counting, and only the midtone, 4/8, is free.

That is proved in a paragraph, and it is also checked. A search that allows the second tone a float of two exhausts 5,040 starting parts and finds nothing; one that holds the second tone to three and allows the third tone two exhausts 10,080 partial chains and finds nothing. Both searches finish rather than stop, which is what makes them a proof rather than a failure to look hard enough.

The rate floor, at eight ends #

The float limit translates into a floor on each tone’s interlacing rate — the quantity the pressing model reads, which decides how far a tone sinks.

With the pinned tones at three and the centre at two, the floor is 0.375 at the shoulders and 0.500 at the centre, against 0.250 at the extremes. The ramp’s depth is set by its firmest tone, so no eight-end shading of this shape can sink less than the thickness at 0.25 less the thickness at 0.5: on a sheeting at half a newton, 42.8 micrometres. That was the floor the six-end essay quoted, and it is now the floor of a family that exists.

The floor is reached #

It is reached, and quickly: the search’s first complete chain sits on it.

This chain has a shape the six-end shadings did not. Every one of its five middle tones interlaces at exactly 0.5, so on a sheeting its surface drops the whole 42.8 micrometres between the first tone and the second, lies flat across the middle, and rises again at the end. The six-end valleys had sloping sides — their second and fourth tones sat at one of three heights between the ends and the bottom — and this one is a trough.

In the first hundred thousand chains the search reached, 27,904 sit on the floor. It is not a rare arrangement.

But most chains sink further #

At six ends every even chain sank to one depth, because the float limit pinned the midtone’s rate at its floor. At eight ends the limit on the centre is a float of two, and a float of two does not pin a rate: a tone of four marks floating two can interlace at 0.5, like a 2/2 twill, or more.

So the family spreads. The first hundred thousand chains fall into six classes by the rate of their worst tone:

• 0.500 — 42.8 µm, 27,904 chains, the floor;
• 0.5625 — 50.1 µm, 7,296;
• 0.625 — 56.7 µm, 35,680;
• 0.6875 — 62.6 µm, 8,960;
• 0.750 — 68.1 µm, 18,624;
• 0.8125 — 72.7 µm, 1,536.

The deepest of them floats exactly as the floor does — 7, 3, 3, 2, 3, 3, 7 — and sinks seventy per cent further. At six ends the floats decided the depth. At eight they do not: two shadings indistinguishable by every float they have differ in relief by thirty micrometres, which on a sheeting is about a tenth of the cloth.

A plain centre is allowed at eight ends #

A six-end shading can be even or have a plain centre, not both: taking a part out of a six-end plain weave or adding one always left a float of three, so a plain midtone and floats of two beside it could not coexist.

At eight ends that conflict disappears, because the neighbours float three anyway. A plain centre with threes beside it exists — the search finds one within a thousand nodes.

It is the deepest shading of all. A plain weave interlaces at every intersection, a rate of 1.0, and the ramp sinks 83.6 micrometres — nearly twice the floor. The six-end essay’s choice between an even ramp and a plain centre becomes, at eight ends, a choice between depths inside one family.

Why six was special #

Put the two repeats side by side and six ends turns out to be the odd one.

At six ends, the float limit of two is tight at every middle tone at once: it forces the midtone to its floor rate, so the depth has nowhere to go. At eight ends the tightest limit the second tone allows is three, the third tone inherits it, and the only tone left free is the one whose rate decides the depth. The limit that pinned everything at six pins everything at eight except the one thing that matters for relief.

So the six-end result — one depth for the whole even family — was a property of the order six, not of shadings. The general statement is weaker and truer: an even shading sinks at least as far as its centre’s floor, and whether it sinks further is decided by the centre’s arrangement, not its floats.

Eight ends is the repeat damask is actually woven in #

This would be a curiosity of combinatorics if eight ends were an unusual repeat. It is the usual one. The eight-end satin is the classic damask weave, and a shading changes two things at once began from an eight-end shading built from its cosets, whose floats run 7, 3, 3, 1, 3, 3, 7 — a plain centre with threes beside it, the very pattern the six-end census ruled out.

So the traditional satin shading was already the eight-end family’s deepest member. A tone ramp is a valley, and the satin digs it measured its sag at eighty-four micrometres between its ends and its middle, and the arithmetic here says why nothing gentler could be had without changing the centre: every tone beside it floats three in any even eight-end shading, so the only freedom left is the centre’s arrangement, and the satin spends it on a plain weave.

A designer who wants a shallower ramp at eight ends has exactly one thing to change, and it is the four-mark tone in the middle. Hold it to a float of two, arranged in pairs as a 2/2 twill has them, and the ramp’s floor is 42.8 micrometres on a sheeting — half the satin’s depth, with every other float the same.

Seven greys, and six of them fixed #

A weave is a halftone screen with n greys: an eight-by-eight cell of cloth gives seven tones between its extremes, where the same cell of printer’s dots gives sixty-three. That essay found the missing greys spent on two requirements — that every thread reach both faces, and that every thread carry the same number of marks.

The pinning adds a third requirement’s price, on the greys that are left. Of the seven greys an eight-end shading can show, the two extremes float seven, four more float three, and only the middle one can be finer. The tone scale is fixed by the repeat; the float scale is fixed by the counting; and the one grey in which a weaver still has a choice is the one that decides how deep the cloth sinks.

What the edges do #

A shaded damask sets two of these tones side by side, and a damask’s edge floats further than its figure found that a float can cross the boundary between two tones and run longer there than in either. With every shoulder tone of an eight-end shading already at three, a boundary between two of them has less room to make things worse than a boundary between a satin and its complement does — the floats being joined are threes rather than sevens. Whether that makes an eight-end even shading’s edges quieter than a satin damask’s is a count of boundaries this essay has not run, and it is the practical version of everything above.

How deep, cloth by cloth #

The depths above are on a sheeting. The floor’s own depth on other cloths was already computed, at eight ends as a lower bound because no chain was then known to reach it.

Those eight-end figures are now attained values rather than bounds: 29.1 micrometres on a batiste, 36.2 on a muslin, 42.8 on a sheeting and 45.8 on a duck, for the shallowest chains of the family. Every other chain found sinks further by the pressing difference between its centre’s rate and 0.5.

The model named #

A shading is a chain: eight disjoint permutation matrices added one at a time, so that tone k is the first k of them — a Latin square of order eight whose symbols are read in order, as a tone step does not need a satin established. A float is the longest run of one face along any pick or down any end, cyclically. The interlacing rate is the changes of face per intersection, the one number the pressing reads. The pressing is the same rule that dug the satin’s valley: a thread presses where it turns, the pressure flattens the yarn, and a tone’s thickness follows from its rate; the depth is the thickness at the extremes less the thickness at the worst tone.

What was counted, and how #

The search builds a chain part by part. The first part is fixed only in where the repeat starts — its first pick’s mark on the first end. Each further part is any permutation that uses none of the cells already used, and it is kept only if the tone it completes floats no more than that tone’s bound, checked pick by pick as the tone is built and then end by end.

Three kinds of result come out and they are different kinds of claim. Refusals — no chain with a float of two at the second or third tone — are exhausted searches: 5,040 and 10,080 partial chains, every one abandoned, the walk finishing. Existences — a chain on the floor, a plain-centred chain — are single witnesses, each checked tone by tone. The family’s depths are from a walk stopped after the first 100,000 complete chains in the search’s own order, so they are a lower bound on how many depths the family has, and nothing here claims the deepest found is the deepest there is.

What the search cannot say #

How large the family is. The squares of order eight number about 10²⁰, the float bounds cut that down enormously, and what is left is still far beyond a walk; the first hundred thousand chains took seconds and the walk was nowhere near finishing. The six depths are six that exist, not all there are.

Whether the depths are spread evenly. The walk visits chains in a fixed order, and a fixed order is not a random sample; the counts in each class describe the walk’s first hundred thousand, not the family’s proportions.

And whether a real cloth shows a thirty-micrometre difference. The pressing model is the one every shading essay has used; its depths are its own, and two shadings with identical floats and different depths are a prediction about relief under a raking light that a weaver with a jacquard could check in an afternoon.

Who found it, and when #

Shaded damasks built from satin cosets are centuries old, and a shading changes two things at once began from them. The counting argument for a float floor is the run-length-limited coding bound of the 1960s read onto a thread. The six-end census and its single depth are this collection’s.

That eight ends pins every tone but the centre at three, and that the family therefore spreads in depth rather than collapsing to one, are new here, and they turn the six-end uniqueness from a law into a coincidence of order.

Still open: what decides the centre’s rate #

Every chain in the family has the same floats; what separates the floor from the deepest is how the centre’s four marks sit in each end and pick — in two pairs, as a 2/2 twill has them, or in pairs and singles, which interlace more.

That is a question about the centre alone, and it is small enough to answer completely: the four-mark tones of an eight-end repeat that float two and can be reached from a pinned second and third tone. Which of them a chain can reach, and whether the floor is reached from every starting part or only some, would say whether a designer who wants the shallowest ramp has to choose the first part carefully or can take any and choose the centre.