An even shading cannot keep its surface level

A six-end shading can be even or have a plain centre, not both walked every one of the 1,128,960 ways to split a six-end repeat into the parts a shading is built from, and found 2,816 that hold every tone between the extremes to a float of two — sixty-four distinct shadings once the corner a repeat is started from is set aside. The twill’s is one of them. The other sixty-three have no diagonal and no satin behind them, and nothing about any of them makes a designer reach for it.

That census answered a question about floats and left one about the cloth. A tone ramp is a valley, and the satin digs it: a thread presses on the thread it crosses only where it turns, a firmer tone is pressed harder and finishes thinner, and an eight-end satin shading’s surface sinks 84 micrometres between its ends and its midtone. Built on a twill read one step at a time, the same ramp is level to the micrometre. The sixty-three have the twill’s floats without its diagonal, and whether they keep its level surface was the next thing to ask.

They do not. None of them does, and the answer is sharper than a no: every even six-end shading sinks, and every one sinks by the same amount.

Every even chain, pressed #

The pressing model is the one that dug the satin’s valley, unchanged. Each tone of a chain is a weave; the weave’s interlacing rate sets how many turns press on each crossing; the pressure flattens the yarn to a thickness; and the surface profile along the ramp is that thickness, tone by tone, measured from the palest tone. The cloth is a sheeting at half a newton in the end.

Each of the sixty-four classes admits several orders of its parts that keep every middle tone at a float of two — sixteen, thirty-two or forty-eight of them — and every one of those chains was pressed.

Every one of them sinks to 43.3 micrometres at its midtone, which on a sheeting 0.263 millimetres thick is a sixth of the cloth. The deepest and the shallowest chain of every class sink to exactly that depth. The twill’s own even chain — the cyclic square added in the order 5, 2, 2, 2, 5 rather than one step at a time — sinks with the rest.

What differs between them is only the shape of the valley’s sides. A chain’s second and fourth tones can sit 25.4 micrometres down, 38.0, or all the way at the floor, and those are the only three heights there are.

Why one depth: a float limit is a run-length limit #

The pressing reads one number from a weave, its interlacing rate, so the depth of any ramp is the difference between the thickness at its extremes’ rate and the thickness at its firmest tone’s. The census’s uniformity has to come from the rates, and it does, from an argument short enough to state whole.

Take a tone of k marks in every end and every pick of an n-end repeat, with no face float longer than f. Along a pick, the n − k unmarked places are weft floats on the face, so they come in at least $\lceil (n-k)/f \rceil$ separate runs. Down an end, the k marks are warp floats on the face, so they come in at least $\lceil k/f \rceil$ runs. On a thread that closes on itself every run is bounded by two changes of face, so a pick interlaces at least $2\lceil (n-k)/f \rceil$ times and an end at least $2\lceil k/f \rceil$ times.

That is the rule a recording engineer knows under another name. A run-length-limited code caps how long a signal may stay unchanged, so that a reader keeps its clock, and the cap forces a minimum density of transitions. A float limit is a run-length limit on a thread, and it forces a minimum rate of interlacing in exactly the same way.

At six ends with floats held to two, the midtone of three marks needs at least two runs of marks down every end and two runs of gaps along every pick: an interlacing rate of at least two thirds. The extremes, one mark and five, float five whatever is done and interlace exactly twice per thread: a rate of one third. The census reaches the floor at every tone — no even chain interlaces its midtone more than it must, and none can do less.

So every even ramp runs from a rate of one third at its ends to exactly two thirds at its middle, and its depth is the thickness at one third less the thickness at two thirds, whatever its square. The sixty-four classes differ only in their second and fourth tones, which sit at the floor’s half, at 0.61 or already at two thirds, and that is the whole of the difference between them.

The twill in order is exactly as floaty as level requires #

The same bound says what a level ramp costs, and it prices the twill’s level ramp to the crossing.

A ramp is level only if every tone interlaces at the extremes’ rate of 2/n. That needs $\lceil k/f \rceil$ and $\lceil (n-k)/f \rceil$ both equal to one, so f at least max(k, n − k) at every tone. At six ends that is floats of at least 5, 4, 3, 4, 5.

The twill read one step at a time floats exactly 5, 4, 3, 4, 5. It is not merely a level ramp; it is the level ramp with the shortest floats a level ramp can have, tone by tone. The price of levelness is written into the float profile the construction has always had, and a designer who wanted both a level surface and a shorter float at any tone was asking for something the counting forbids.

At every order the midtone of a level shading floats half the repeat, rounded up. An even shading holds its floats to two at every order, so the two properties coincide only at four ends, where half the repeat is two — and a four-end shading is the one repeat at which even and level are the same thing.

The shoulders are all that is left to choose #

With the depth fixed, what a designer choosing among the sixty-four is actually choosing is the shape of the valley’s sides.

Five shapes occur. The steepest, reached by nine classes including the twill’s, drops to the floor at the second tone and stays there through the fourth: a flat-bottomed valley with a single step of 43.3 micrometres at each end. The gentlest, reached by eleven classes on each side, holds one shoulder at a rate of one half, 25.4 micrometres down, so that side descends in two steps of about 25 and 18. The commonest, twenty-four on each side, holds one shoulder at 0.61, 38.0 micrometres down.

The flat-bottomed classes are the ones a designer would take for the most even of the even, because every middle tone is at once the shortest-floating and the firmest the repeat allows. They are also the ones whose relief is least spread out: all of it sits in two boundaries, one either side of the dark half of the ramp, where a shouldered chain would have split each into two lines of about half the size.

That makes the choice among even shadings a choice about where the lines in the surface fall. A figured cloth has a step in its surface wherever two weaves of different interlacing rate meet, and a shading is a design made of such boundaries. A flat-bottomed valley puts all its relief into two boundaries, one at each end of the dark tones; a shouldered one splits it between two smaller boundaries on one side. Neither removes any of it.

How deep the valley is on other cloths #

The depth depends on the cloth, because the pressing reads a construction — sett, count, tension — as well as a rate. It does not depend on the shading, which is the point of having found it as a pair of rates.

On every cloth tried the valley is between a tenth and a fifth of the cloth’s own thickness: 29.7 micrometres on a batiste, a fifth of it; 51.7 on a duck, a tenth. The heavier the cloth the deeper the valley and the smaller its share, because a thick cloth’s thickness is mostly thread the pressing cannot flatten.

At eight ends no census has been run, but the bound gives a floor. An even eight-end midtone interlaces at least half as often as it has crossings, against a quarter at the extremes, so every even eight-end shading sinks at least 42.8 micrometres on the sheeting. The satin spread chain that dug the first valley sank 84, because its midtone is a plain weave at the top rate there is. An even eight-end shading, if one exists that reaches the floor, would sink half as far.

What the valley is not #

The valley is not a flaw in the sixty-three shadings that the twill avoids. The twill’s own even chain sinks by the same depth, and the twill avoids it only when it is read in the order that floats five, four, three, four, five. Levelness belongs to the order, not to the square, which is the same correction the crown line’s identity needed when it was traced back from the chain to the move.

Nor is it a property of the firmness averaged over a cloth. A cloth slips at its least-interlaced thread, and a thread’s grip is its own crossings; the pressing here is the same count read as a force on the crossed thread, averaged over a tone region, and the region is what carries the relief.

Nor is it large enough to see in flat light. Forty micrometres is a hair’s width. It is the kind of relief a raking beam reveals as a line where two tones meet, and that account found a figure’s shine does more to make it visible than its step. What the census adds is that no even ramp can remove its lines by choice of arrangement — it can only choose where they are.

Four ends is the one repeat where both are free #

The level-float line and the even line meet at four ends, and it is worth saying what that means in cloth, because it is the one place the conflict above does not exist.

A four-end shading has three tones, one, two and three marks in four. The extremes interlace at a rate of one half — two changes of face on a four-crossing thread — and the midtone, two marks in four held to a float of two, needs $\lceil 2/2 \rceil$ runs of marks and $\lceil 2/2 \rceil$ runs of gaps: also one half. So an even four-end shading is automatically level, and a level one automatically even. The twill in order floats three, two, three, and so does every even four-end chain.

That is also why nothing at four ends ever looked like a valley. A tone step does not need a satin counted twenty-four four-end decompositions and found their chains, and a pressing model run over them would have found every surface flat — not because the chains are well chosen but because a four-end repeat has no room for a float limit to bind. From five ends up the two budgets separate, and they separate further with every end added: at sixteen, a level ramp floats eight where an even one floats two.

The practical order of that finding is the reverse of how shadings are usually designed. Shading is a technique of fine repeats — five, eight, twelve ends — precisely because a fine repeat gives more tones, and a weave’s tone count rises with its repeat. Every end added for another tone widens the gap between the float a level surface needs and the float a sound cloth tolerates. More tones and a flat surface are bought from the same ends, and four ends — one tone between the extremes — is the only place they are not in competition.

A weave’s tone scale and its relief are separate budgets #

A weave is a halftone screen with n greys, and a printer’s screen has no surface to speak of. A weave does, and the account here shows the two scales are governed by different constraints on the same matrix. The tone scale is set by marks per thread, the relief by runs per thread, and a shading that is even in one of them — every tone at the shortest floats — is forced to be uneven in the other.

That is why no specification of a shading is complete in tone alone. A designer who asks for an even ramp has, without saying so, asked for a valley of a known depth, and a designer who asks for a level ramp has asked for floats of half the repeat at the midtone. Designing to a float limit treats the float as the cloth’s safety margin; here it is also the cloth’s flatness, and the two pull in opposite directions at every order above four.

What was computed, and how #

The even family is the census’s: every Latin square of order six with its first row in order whose best chain holds the three middle tones to a longest float of two, grouped into classes by cyclic shift and reversal of picks and ends. For each class, every ordering of the six parts was tried and kept if its three middle tones floated at most two. Each tone was pressed by the site’s pressing rule at the sheeting’s construction and half a newton — interlacings per unit area times the contact force at a crossing, flattened to the thickness that force produces — with the thickness cached by interlacing rate, since the rate is all the rule reads.

The float floor was confirmed to be reached, and not undercut, at every middle tone of the census; every one of the sixty-four classes was confirmed to sink, and every chain of every class to reach the same midtone depth; the twill in order was confirmed level to the micrometre, and its floats confirmed equal to the least a level ramp needs at every tone; and the midtone float a level shading needs was confirmed to be half the repeat, rounded up, at every order from four to sixteen.

Where the pressing model stops #

The pressing is local. Each tone is pressed as though it filled the cloth, and a real tone region is a block with neighbours; at its boundary the two thicknesses meet over a width of a few threads that nothing here models. The depths are the interiors’ difference, not the shape of the step.

Only the rate is read. Two weaves at one interlacing rate press alike in this model whether their interlacings are clustered or spread, and a real cloth’s thickness may depend on the arrangement too. The census’s uniformity is uniformity of rate, and it is exact; the uniformity of thickness follows only through the model.

Nothing is finished. A calender or a press flattens a cloth’s high points first, and could take much of a forty-micrometre valley out or print it more sharply, and neither is computed.

Still open: whether an eight-end even shading reaches the floor #

The six-end census found every even chain at the float floor’s rates, which is what made the depth unique. At eight ends the floor says every even shading sinks at least 42.8 micrometres on a sheeting, and says nothing about whether any reaches that floor or whether they spread out above it. The Latin squares of order eight number about 2.7 quadrillion with their first row in order — 535,281,401,856 even with the first column fixed as well — too many to walk one at a time. But the floor is a statement about runs per thread and the census only needs the squares whose middle tones stay at a float of two, and a search that builds squares row by row and abandons any partial square whose partial tones already float three might reach the eight-end family without walking the rest. Whether an eight-end even ramp can be as shallow as the floor, and how many distinct depths the family has, is that search’s question.

Who found it, and when #

Shaded damasks built from satin cosets are old Jacquard practice, and run-length-limited codes are a standard tool of digital recording. The census of six-end decompositions and its even family were counted here; pressing every even chain, finding that all of them sink to one depth, deriving the float floor on interlacing rate that forces it, and pricing a level shading’s floats as half its repeat at the midtone, was done here.