A damask's edge floats further than its figure

A damask is its own complement lists what figure and ground share, and every entry on the list is exact. One satin, used face up and face down. The same shafts, the same threading, the same number of interlacings. And the same longest float: seven picks on an eight-end repeat, whichever of the two areas is measured.

Every entry is a property of an area. A damask is not only two areas; it is two areas and the line where they meet, and a float has no obligation to stop at a line drawn on point paper. A warp end on the face at the last pick of the figure can stay on the face into the first pick of the ground, and whether it does is decided by neither satin on its own.

Along the edge of a damask whose ground is the exact complement of its figure, it does — on one end in every eight, and the float is eight picks long. Nothing inside either area floats further than seven.

A float that belongs to neither area #

The float that crosses an edge is an ordinary object once it is looked for, and the first thing to settle is which floats can do it.

An edge running across the cloth lies between two picks. The only threads that pass from one side of it to the other are warp ends, so the only float that can cross it is a warp float — a run of consecutive picks over which one end stays on the face, begun in one area and finished in the other. An edge running up the cloth lies between two ends, and by the same reasoning the only float that can cross it is a weft float.

Nothing else crosses anything. A weft float beside a horizontal edge lies wholly within one pick and therefore wholly within one area, and a warp float beside a vertical edge lies wholly within one end. So each direction of edge has exactly one kind of float to answer for, and a figure, which has a top, a bottom and two sides, answers for both.

Where the eighth pick comes from #

The mechanism is short, and it is worth following on one end rather than stating as a count.

In the warp-faced figure each end goes under the weft once in eight picks and lies on the face for the other seven. Take the end whose next interlacing, had the figure continued, would have fallen on the first pick below the edge. Above the edge it has been on the face for seven picks, since its previous interlacing a whole repeat earlier.

Below the edge the figure has stopped and the ground has begun — and the ground is the complement. Exactly where the figure would have taken this end under a pick, the ground puts it over. The interlacing that was due does not happen, the end stays on the face for an eighth pick, and only then does the ground’s own pattern take it under.

Every end of a satin interlaces at a different pick, so at any one edge exactly one end in eight is in that position. The float is eight on that end and shorter on the other seven, and the same is true of the bottom edge of the figure with a different end.

The sides of the figure behave the same way with the roles exchanged. A pick in the warp-faced figure appears on the face at one end in eight; in the weft-faced ground it is on the face at seven. The pick whose single appearance in the figure falls at the last end before the edge finds the ground keeping it on the face for the next seven, and the float along that pick is eight ends long.

One pick along, and the edge closes #

The complement is not the only weft-faced satin available for the ground. There are eight: the same move, started at any of the eight picks. The complement is the one that puts its single warp mark per end exactly where the figure has its single weft mark. Every other start puts the ground’s mark on one of the figure’s seven.

That is the whole difference. Wherever the figure takes an end under a pick, a ground started one pick along takes it under too, so the interlacing that was due at the edge happens and no float runs on past it. The longest warp float across the edge is seven — the figure’s own — and the longest weft float along the sides is five.

So of the eight grounds that would each be described as the figure’s weft-faced counterpart, one lengthens a float at the edge and seven do not, and the one that does is the one a description in terms of complements names.

Some starting points bind the edge tighter than the cloth #

The sweep has a second result, and it runs the other way.

Starting the ground halfway along — four picks — gives an edge whose longest warp float across it is four and whose longest weft float along it is four. That is shorter than anything inside either area, which floats seven. The edge of such a figure is the firmest line in the cloth.

The pattern is symmetric: a ground started two picks along and one started six picks along give the same pair of numbers, as do three and five, and one and seven. Four is the middle of that symmetry and the tightest.

That bears on a practice the damask essay records: outlines in figured cloth are frequently reinforced with a separate stitching weave, to sharpen the edge. A figure drawn against a ground started halfway along has a reinforced edge already, at no cost in threading and no change in either satin — a quarter of the cross-edge float of the complement, bought by starting the ground somewhere else. Whether any designer places the ground for this reason is not something a matrix can report; that the choice exists, and that the complement is the worst of the eight, is.

What the shading rule was protecting #

The rule that tones must nest is not new to this argument. A shading changes two things at once required it of every chain of tone steps: each step must be the previous one plus a part, because otherwise crossing the edge between two tones “turns some intersections from warp-up to weft-up as well as the other way — and that shows as a line the design did not draw.”

The line was left undescribed. It has a length, and the length is a float.

A damask’s figure and ground are the two ends of a shading on the same repeat — the tone of seven marks per end and the tone of one. Built from a chain, the paler tone’s single coset is one of the darker tone’s seven, and the pair nests. The exact complement is not a chain: its one coset is precisely the one the figure leaves out. So the damask essay’s construction and the shading essays’ rule disagree about the one thing that decides an edge, and a figure and ground drawn from one chain are any of the seven grounds that do not lengthen anything.

Why nesting is enough, at an edge of any shape #

The eight grounds are a sweep. The reason seven of them behave is not, and it is short enough to give whole.

Call the paler tone’s warp marks P and the darker tone’s D, and suppose every mark of P is also a mark of D. Every intersection of a design is taken from one tone or the other. So wherever the edges of the design run, its warp marks include all of P and lie entirely inside D.

A warp float in the design is a run of consecutive intersections along one end, every one of them a warp mark. Each is a mark of D, so the same run is a run of D’s marks and can be no longer than D’s longest warp float. A weft float in the design is a run with no warp marks in it. None of those intersections is a mark of P, so the same run is a weft float of the paler tone and can be no longer than P’s longest.

The darker tone bounds every warp float and the paler tone bounds every weft float, at every edge, and the shape of the edge never enters. No enumeration is needed, which is the reason to state the argument: a census of edges would be a census of shapes, and there is no end of those.

The same argument covers the shading essays’ other chains. A tone step does not need a satin builds chains out of Latin squares rather than satin cosets, and nothing above mentioned a satin: it used only that each tone is a set of marks and that one set lies inside the next. Any nested chain, from any decomposition, has edges that float no further than its tones.

A design of any shape, drawn twice #

A straight edge is the simplest case, and the argument is about every case, so it is worth seeing on a design that has edges in both directions and a corner.

The three tones on the left carry one, three and six marks per end, and the design’s longest warp and weft floats are six and seven — the longest the three tones have between them. On the right the tones carry the same one, three and six marks per end and give the same average tone in every block, and the design floats nine in the weft against the seven its tones allow.

A reader looking at the two drawings from a distance sees the same design. The tone of every block is identical, and the difference is entirely in which intersections carry the marks — which is exactly the kind of difference the halftone argument found a shading’s tone count cannot see.

Every pair of tone steps on eight ends #

The argument says nested pairs never lengthen a float. It does not say how often a pair that does not nest does, and that is a count.

An eight-end satin with move three has 254 tone steps — every union of one to seven of its eight cosets — and 32,131 unordered pairs of them. Each pair was put either side of a straight edge, in both orders, running across the cloth and running along it, and the longest crossing float compared with the longest float either tone has on its own.

The nested row is a check on the argument and comes out as the argument says: 5,796 pairs, not one of which lengthens a float. The interesting row is the second. Of the 26,335 pairs that do not nest, 25,327 lengthen a float somewhere along their edge — ninety-six per cent.

So the rule is not merely sufficient. It is very nearly what it takes. A designer who ignores nesting and picks tones independently, each for its own lustre or its own float, has about a one-in-twenty-six chance that a given pair of them meets cleanly. The 1,008 that do are not reached by the argument above, and the census does not say what they have in common; it says only that they are rare.

How much further an edge floats #

Lengthened is one question. By how much is another, and it is the one that decides whether anybody would notice.

Four pairs in five lengthen the float by one intersection or two, and that is the typical fault — a float a little longer than its neighbours along a line. At the other extreme sixteen pairs double it, and every one of the sixteen is the same situation. Half of them are two weft-faced satins started three picks apart, each floating seven ends on its own: at a side edge the pick that is on the face at the last end of one region is on the face at the first ends of the other, and the float runs fourteen. The other half are the two warp-faced satins, doing the same with a warp float of fourteen across a top or bottom edge.

Those pairs are worth naming because they are the most innocent-looking in the census. Both regions are the palest tone there is, or both the darkest; both are the same weave; and a design that used one of them in one place and the other nearby would be described as using one tone twice.

A damask is always the edge case #

The complementary pairs are the smallest row in the census and the only one that lengthens a float every time. A damask’s figure and ground are such a pair, and the damask sweep says what happens across orders.

At five ends, at seven, at eight and at twelve the result has the same form. Exactly one of the n places the ground can start lengthens a float at the edge, it is the exact complement, and it lengthens the warp float and the weft float both by exactly one — to n, against the n − 1 that either satin floats inside. At twelve ends the ground started halfway along floats six, half of what the complement does.

The excess of one is the smallest excess the census contains, and it would be easy to dismiss. It is worth one more section, because one pick is precisely what the damask’s own order was chosen against.

One pick is the one the float limit was drawn at #

The damask essay derived why the classical damask is an eight-end satin. At twenty-four picks to the centimetre a pick spacing is 417 micrometres, an eight-end satin floats 2.9 millimetres, and the rule figured weaving works to — a float of about three millimetres on a cloth that will be used — lands between eight and twelve. Eight is the largest order whose float fits under the limit.

The edge float of an exact-complement damask is eight picks rather than seven, and at the same sett that is 3.3 millimetres. It is over the limit that chose the order, along every top and bottom edge of the figure, on one end in eight. At twelve ends and thirty-six picks to the centimetre — the fine damask the same essay offers as the way past eight — the interior float is 3.1 millimetres and the edge float 3.3.

Designing to a float limit treats the limit as a property of a weave, and for a single weave that is what it is. A figured cloth has a longest float the weaves do not, and the edges are where it lives. They are also where a figured cloth wears first whenever a float stands proud, which a longer float on an outline does by exactly the amount that makes an outline visible.

The origin, characterised for one pair #

What else the relative origin decides swept every position of a ground weave against a figure weave and found that the longest float moves — from four to five on an eight-end satin figured on its own reverse at small blocks — and left the question of which origins give which float as a list rather than a structure.

For regions a whole repeat deep, the nesting argument supplies the structure for this pair. The eight origins split into the complement, which is the only one that lengthens a float; the seven that nest, which cannot; and inside those seven a symmetry that pairs a ground started s picks along with one started eight picks less, down to the tightest edge at half a repeat.

That is a characterisation rather than a count, and it is exactly what the sweep could not give: a reason the one bad origin is bad, which is a sentence about subsets rather than a row in a table. What it does not cover is the smaller blocks that essay worked at, where a region is less than a repeat and a float can be cut short by the next edge before it reaches its length.

What point paper cannot show about an edge #

Everything above is drawn one intersection wide, and a woven edge is not.

The two tones either side of it interlace at different rates, so they take up yarn differently and press their crossings differently, and a shaded ramp’s tones stand at different thicknesses for that reason. An edge between them is a place where the cloth changes from one thickness to another, and that change is spread over several threads rather than taken in one step. A float of eight on the drawing is a length of yarn lying across that transition.

Nothing here says how that float lies, or whether an eye finds it. A float one intersection longer than its neighbours may vanish into the lustre of the regions either side, or it may catch the light along the whole outline — and a figure shows by its shine, which is to say by exactly the property a longer float changes. The counts are exact and the visibility is not computed.

Nor does the drawing say anything about the weft’s own path at the sides, where a float of eight ends crosses between regions whose picks are packed to different densities.

Where the argument comes from #

That shaded tones are built by adding marks to marks is old practice in figured weaving, and so is outlining a figure with its own binding. The observation that a nested pair of tones bounds every float at every edge — the darker tone bounding the warp and the paler tone the weft — is an elementary fact about sets and runs, and it does not appear in the weaving sources consulted here. The damask result follows from it immediately once a damask is recognised as a pair of tones.

The census takes every union of one to seven cosets of the eight-end satin with move three, pairs them all, and for each pair walks outward from a straight edge between two repeats of each tone, thread by thread, in both orders and both directions, counting how far one state continues on each side. A crossing float is at most two repeats less two intersections long, so a region two repeats deep holds any float whole. The damask sweep is the same measurement at every starting pick of a single-coset ground against the seven-coset figure, at five, seven, eight and twelve ends.

Still open: how wide an edge is in cloth #

Every result here is about a figure drawn on point paper and read as a matrix. The next question is the one a figured cloth’s step in its surface raised and could not finish: two tones meeting in a woven cloth take up yarn at different rates, so an edge is a ridge or a valley several threads wide, and a float of eight lies across it rather than beside it. How wide, and whether the ground started halfway along shortens the ridge as it shortens the float, needs a model of take-up across a boundary that nothing here has.

Sideways, the same edge-float question has a much older form in a twill that changes direction. A broken twill and a herringbone are two ways of reversing a diagonal, and one of them leaves two neighbouring ends doing exactly the same thing along the reversal — a float at a join, arrived at from a completely different construction.