A figured cloth has a step in its surface

A damask is one cloth. One warp, one weft, one sett, one set of threads throughout, and a pattern made entirely by changing which weave is used where. That is what makes it the pattern field’s cleanest object: nothing about the figure differs from the ground except the matrix.

Hold a damask up to a raking light and the pattern is visible in relief. It is usually explained by lustre — a satin figure reflects where a sateen ground scatters — and that explanation is right and is not the whole of it. The cloth is genuinely not flat.

Where the pressing goes #

A thread presses on the thread it crosses where it turns. Over a float it lies on the surface, passes over threads it is not wrapped around, and presses on nothing — which is the correction the grip ladder made this rung and is a property of the draft with no geometry in it.

So the pressing a region of cloth receives per unit area is the contact force at one turn, times the turns per unit area:

$$P = N \times \frac{\text{interlacings in the repeat}}{\text{area of the repeat in cloth}}$$

and the second factor is a count off the matrix, exactly.

A five-end satin turns two fifths as often as a plain weave. So at the same warp tension, in the same cloth, a satin region is pressed two fifths as hard as a plain one — and there is nothing about the yarn, the sett or the finish in that ratio.

Less pressed means less flat means thicker #

A crossing flattens because something pressed it, and how flat it gets is a monotone function of the force. Half the pressing gives a rounder section, a rounder section is thicker through the cloth, and the cloth is thicker where the section is.

Putting the two together gives the step.

Fifty micrometres is a twentieth of a millimetre, on a cloth about a quarter of a millimetre thick — so the figure stands about a fifth of a fabric thickness proud of its ground. That is far more than the eye needs. Raking light will show a step of a few micrometres on a matt surface, and this is ten times that.

Which explains the raking light without lustre #

The usual explanation is a reflectance difference: a warp-faced satin is bright and a weft-faced sateen is not, so the pattern appears when the light comes from a direction that favours one. That is real, it is why a damask reverses when it is turned round, and it is why a damask in a single colour works at all.

But it predicts that the pattern disappears at normal incidence and in flat light, and it does not entirely. A damask examined in diffuse light, or felt with a fingertip, still has its pattern — and a fingertip does not read reflectance.

The step is a second, independent channel and it does not turn off. That is worth having because it explains a practical fact: a damask holds its pattern after dyeing, after a heavy finish, and in a colour where the lustre difference is small — because the relief is a property of the construction rather than of the surface.

There is a third channel that the step opens and it is worth naming, because it is the one a manufacturer meets. A cloth that is thicker in some regions than others does not roll flat, does not lie flat under a cutting knife, and does not calender evenly — a nip closes onto the high regions first. So a heavily figured cloth with a large step is harder to finish than a plain one of the same construction, and the difficulty is proportional to the same ratio the pattern is made of.

And it is the same fact that makes a figure cost the loom #

What a figure costs the loom found that a figured cloth’s two weaves demand different amounts of warp, because they interlace different numbers of times and each interlacing costs thread. That is a take-up problem: the figure and the ground draw down the beam at different rates and the cloth cockles or needs two beams.

This rung is the same count read as a pressure rather than as a length. The interlacing rate decides both how much thread a region eats and how hard it presses, so a figure that is easy on the loom is a figure with a small step — and the designer’s rule about keeping figure and ground close in take-up is simultaneously a rule about keeping the surface flat.

That is not how the rule is taught, and the connection is exact rather than analogical: it is one number doing both jobs.

The size, and what sets it #

Fifty-five micrometres for a five-end satin figure on a plain ground in a sheeting, at half a newton of thread tension. It is worth taking that number apart, because three quite different things go into it and only one of them is the pattern.

The interlacing ratio is 0.4 and is the matrix’s. It is exact, it has no units, and it is the only part of the answer that would be the same in any cloth.

The contact force is 1.35 N in a sheeting at ordinary tension, and it carries a thread tension somebody chose and a weave angle from a geometry. Doubling the tension doubles the pressing in both regions and does not double the step, because the flattening’s response to pressure is a saturating curve — so a cloth woven at high tension has a smaller relative step than one woven slack.

The compression response carries the fitted transverse modulus, and a limper yarn gives a larger step because both regions flatten further and the difference between them opens.

So the step is a pattern property multiplied by two fabric properties, and only the first is under the designer’s control at the drawing board. A designer choosing a figure chooses the ratio and not the size, which is why the same design in two cloths reads quite differently in relief and why relief is a mill’s problem rather than a designer’s.

What was counted, and how #

The interlacing counts are enumerated from the matrices by the function this site has used since its first essays, and the turns per unit area follow from the repeat’s extent in cloth.

The step is asserted as two relations rather than as a size. A figure with a lower interlacing rate than its ground must be pressed less and must stand proud of it — the direction — and among figures, a lower rate must give a larger step — the ordering. A version of this that named a step in micrometres would be an assertion about a sheeting at half a newton of thread tension.

The contact force comes from the weave angle Peirce’s geometry solves for at the cloth’s construction, and the flattening from the compression energy the mechanics field built this rung. Both carry the fitted transverse modulus that ladder is honest about, so the sizes here are quoted to two figures and the ordering is what they are for.

And a damask is not one of the pairs. The three pairs computed put a figure on a plain ground, because a satin and its own complement have identical interlacing counts and would give a step of exactly zero. That is a real prediction and it is a sharp one: a true damask, figure and ground being complementary weaves, should have no step at all and its pattern should be pure lustre. A satin figure on a plain ground should have both.

What a weaver would do with it #

The ratio is the designer’s and the size is the mill’s, and separating them makes two practical statements available that were not.

A design can be checked for relief before a cloth exists. The interlacing rates of the figure and the ground come off two matrices, so the ratio between them is known at the drawing board. A designer who wants a flat cloth picks weaves of similar interlacing rate — which is the same rule as keeping their take-up similar, and is now known to be the same rule rather than a second one that happens to agree.

And a design can be checked for the opposite. A relief is sometimes the point: a self-coloured damask, a figured napkin, a matelassé effect got without a second warp. There the rule inverts and the designer wants the ratio as far from one as the loom allows — which means a long-float figure on a plain ground rather than a satin on its own sateen, because a satin and its complement interlace identically and give a step of exactly nothing.

That last pairing is worth stating as the sharp form of the result. The classical damask, figure and ground being complementary weaves, is the one figured cloth with no step at all. Its pattern is pure lustre. Anything else — a satin on a plain, a twill on a hopsack, a figure on any ground that interlaces differently — has both channels, and the second is the one that survives dyeing, finishing and being looked at in flat light.

None of that needed a new measurement. It needed the interlacing count, which this collection has computed since its first essays, to be read as a pressure instead of as a length.

The step-free class is larger than the damask #

The damask is picked out above as the one figured cloth with no step, on the ground that a satin and its own complement interlace identically. That is right and it is not the general condition, which is worth stating because the general condition is both larger and easier to check.

The step is zero exactly when the two weaves have equal interlacing rates. Complementary weaves are one way to get there — turning a draft over exchanges warp-up for weft-up and leaves every turn where it was — and they are not the only way.

Read against the weaves this collection names, the equal-rate classes sort themselves at once. Four of them interlace at one half: the 2/2 twill, the 3/1 twill, the 2/2 basket and the 2/2 herringbone. Any figure drawn from that set on any ground drawn from it has no step at all, and the four look nothing like each other — a diagonal, a warp-faced diagonal, a chequer and a chevron.

So a designer wanting a pure-lustre pattern has a much wider vocabulary than the damask’s. A 3/1 twill figure on a 2/2 basket ground is flat, patterns strongly, and is not a damask by any definition anybody uses.

Two things follow that the complementary reading does not reach.

The step-free class is exactly the one-beam class. Equal interlacing rates means equal take-up, which is the condition for weaving figure and ground from one warp beam — so the weaves that lie flat are precisely the weaves that do not need two beams, and the mill’s constraint and the designer’s are one constraint rather than two that happen to agree. That is the strongest form of this rung’s central connection.

And the class is checkable before anything is drawn. An interlacing rate is a count off a matrix, so a designer picking a figure and a ground can compute the step’s ratio in a moment and know both whether the cloth will lie flat and whether it will weave on one beam. Neither question needs a yarn, a sett or a trial length.

The caution is that equal rates give a step of zero only in this model, where the pressing is a rate times a uniform contact force. Two weaves at the same rate can still have different weave angles — a 3/1 twill’s turn is not a basket’s — so their contact forces differ a little and the step is small rather than exactly nought. The complementary pair is the only case where it is exactly nought, because there the two weaves are the same weave seen from two sides and every geometric quantity agrees. That is why the damask keeps its special place: the wider class is flat and the damask is flat by construction.

Where the model stops #

Both regions are given a plain weave’s weave angle, which is the limitation every Peirce argument on this site carries and here it bites in a known direction. A satin’s crimp is genuinely smaller than a plain weave’s at the same construction, so its turns are gentler, so its contact force at each turn is lower too — and the pressing ratio is therefore smaller than the interlacing ratio alone says. The real step is larger than these figures, by an amount not computed.

The boundary is not modelled at all. A step needs somewhere to happen, and what happens at the edge of a block is a transition over some number of threads with a slope to it. Nothing here computes the width of that transition, which is what decides whether the figure reads as a crisp relief or a soft one, and it is the quantity a designer would most want.

The two regions are assumed to be at the same tension. They are not: a region that takes up more warp is under a different tension from its neighbour on the same beam, which is exactly the take-up problem, and that difference feeds back into the contact force. The model treats the tension as uniform and the interlacing rate as the only variable.

A step is not the only thing a differential pressing does. The two regions also end up with different covers, different air permeabilities and different handles, because all four follow from the same flattening. A damask is therefore inhomogeneous in more ways than its pattern, and the others have never been measured because nobody expected them.

And the step is an elastic step, computed while the cloth is being pressed. A finished damask has been through a calender and a wash, and how much of a differential flattening survives needs the plasticity nobody here has.

The generalisation #

When a quantity is delivered at discrete points, a region’s share of it is set by how many points it has — and two regions of one object can receive different treatments from a uniform environment.

The environment here is a uniform warp tension across the whole cloth. It is delivered to the fabric only at interlacings, so a region with fewer interlacings receives less of it, and a cloth that is uniform in every specified respect ends up non-uniform in a respect nobody specified.

The same shape recurs wherever a load is transmitted through discrete contacts into a continuum: a bolted joint where one flange has more bolts than another, a fibre-reinforced panel where the tow crossings differ in density between zones, a printed circuit where thermal paths are unevenly distributed. In each case the setting is uniform and the treatment is not, and the ratio is a count.

The second lesson is narrower and is about explanations that are sufficient. Lustre explains a damask’s pattern and is sufficient, which is why nobody looks further — and a sufficient explanation is the hardest kind to add to, because there is no residual to be curious about. It took a quantity computed for another purpose entirely to notice that a second channel was there.

Who found it, and when #

That figured cloths are not perfectly flat is known to anybody who has handled one, and the relief of a damask is remarked on in the weaving literature as an aesthetic property.

The take-up difference between figure and ground is a standard weaving problem with a standard answer — match the weaves, or use two beams — and is old.

The compression side is this site’s, from the mechanics field this rung, and the connection between it and the take-up rule appears to be nobody’s stated result. It follows in one line once the pressing is written as a rate times a force, and the line is only available once a thread’s flattening has an energy behind it.

Where the ladder goes next #

Along the blocks ladder, the boundary is the gap: everything here is about the interior of two regions, and what a designer wants to know is what the edge between them does.

Sideways, the prediction that a true damask has no step and a satin-on-plain figure has a large one is testable with a micrometer and a ruler, and is the cheapest thing this rung has produced that somebody could check.

The same interlacing count decides how far each region frays, which means the figure and the ground of a figured cloth have different cut-edge behaviour — and a cut through a damask crosses both.